Uniformization for algebraic curves (in Greek)

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Uniformization
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(TˆwkcSb0‰VrŠuŒ‹‡ep†v‚„u{‚€y Riemann
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vkwBc‘b‰pY’q$rŠ‚‰uwB…v†s‡ˆY‚‰uy‘’‚zt‰gs“fn Riemann
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Galois
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An expert, is a man who has made all the mistakes, which can be made, in a
very narrow ﬁeld.
Bohr, Niels Henrik David (1885-1962).
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jƒ‡{gdywBgsidqprƒtvuwBŽsyRgprŠ‚‰uwB—p†puˆ‡Žsy p : (Y, y) → (X, x)
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p−1
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p‡u{g%xž†pg G
wBepidqdjdjƒgkcR›au{gptfqprƒs††p…Šqsjƒ‚ t‰…%¡T‚zsV™fgdjdg bsc´Yb$wBgp¡‡sVywBgƒuGt‰… ¡T‚€jƒ‚ziŠusV—f‚€y¡T‚€sV™dgdjdg¤tdgdyh¡G‚zyV™v‡{gsy Galois
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Galois
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p‡‚ž”ƒ™vuwBx€y~wBgsjdrƒnsid‚€y jsrŠ…Š™f‚‰‡††pg t‰…D—f…Šnsjƒ‚ ˆ‡gv†Exz†vg wBgv¡Ggs™fe gv†pgpiƒqptvuwB— gp†vtfuwB‚‰‡Œjd‚ž†p…—ƒrƒyVy‚‰‡†pgdu‡…Buk‚zrBu{‹Tev†p‚‰u{‚€y Riemann
cV‘‚¦t„g‚‚€™dpYgsid‚‰‡Œgitfgsygv†pepiƒqsˆ‡gsy$wB‚€™f—v‡{f…Šqsjƒ‚$tfgv†t„…drBuwBŽ¡Gx€gsˆ‡g¥t‰…Šq~gv†vtvuwB‚‰u{jƒxž†p…ƒq¤‚ž†vs®jƒ‚itfgv†¤epidpY‚ž”Š™dg¥tdgp†¥wŠgp¡T…siƒuwBŽ$8jsrŠ…Š™d…Šndjd‚†pg—f…Šqpiƒx%T…ƒqdjd‚$rƒep†fy¬gprŠ——fuŒes‹T…ƒ™fg‚ˆksVjƒgpt‰g
cV‘‚ztfe£gsrƒ—jŠ‡{gi†vnd“fg‚ˆYtfuyXgv†pgpiƒqptvuwBx€y$ˆYq
†pgd™„tfŽsˆ‡‚‰uy¦gsrƒ—%tfgv†£jŠup‡gs—vuwBŽigv†pesidqsˆ‡g©§s”didx€rƒ…Šqsjƒ‚'tdgp†£jƒupYgd—vuwBŽi—f…ƒjƒŽ‚t€y††£ˆYqdjsrŠgppks††‚€rŠu{‹Tgv†p‚‰u{s†† Riemann
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iƒqsjƒjdg'!frŠ‚zt„s††vt‰gdy(£wBeprŠ…Bu{garƒ‚€rƒ‚€™fgdˆYjƒxž†pgˆYgsjƒ‚‰‡{g)8ˆYgdjd‚‰‡Œg¥—vu{gpwŠided—dyVˆYgsy wBgv¡‡sVyiwŠgƒutfgv†©‚‰uwB—p†vg©gdqptds††~jƒx€ˆky tfgsygp†vgsidqstfuwBŽdy‚gsrƒ‚‰u{wŠ—p†suˆ‡gsysc¤©rŠ…ƒ™f‚‰‡—ŠjdyVy%†vg‘p‡‡†p‚‰uwBgdut„…¤gv†vtv‡ˆktf™d…Š‹T…TcV$rŠ—¥jƒ‡Œg‚wBgsidqprƒtvuwBŽ£gprŠ‚‰uwB—p†puˆ‡g‚jdrƒ…Š™f‚‰‡k†pg‚rŠ™f…ƒwŠn%‡‚„uYjŠ‡{gagv†pgpiƒqptvuwBŽgsrƒ‚„uwB—v†suˆYgm‚zrBu{‹‡gp†p‚‰us†† Riemann
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c£‘‚ztfe …Š—fgpp‡…ƒndjdgdˆktf‚iˆkt„…©rƒ—Š™vuˆYjƒg wkc±¤©§6—ƒrƒ…Šq¤wŠep¡T‚‚gv†peppkyRp‡…rŠ…piƒqps††pqsjƒ… F
—fns…©jƒ‚zt‰gv”ƒidgst„s††¥ˆkt„…v† C
‚zrŠepp‡‚‰u jŠ‡{g¥wBgpiƒqprƒtfu{wŠŽ¤gprŠ‚‰uwB—v†suˆYg¨§Tgsrƒ—mt‰…gsispY‚ž”Š™vuwB—¤ˆYnp†p…pid…mjƒgs—f‚ž†suˆYjƒ…Šn¥t‰…Šq©§Gˆkt„… C
gs‹Tgdu{™ds††vt‰gsy‚xž†pg¥rŠ‚zrŠ‚€™fgsˆYjƒxž†p…~ˆYnv†p…sid…Tc²6†fyV™v‡{f…p†ft„gsy£—stfu6wBgpidqsrdtvuwBx€ygprŠ‚‰uwB…p†p‡ˆ‡‚‰uyh‚zrƒespY…Šqv†gv†pgpiƒqptvuwBx€ywBgdu proper
ˆYqp†vgd™
tfŽsˆ‡‚‰uy¤jƒ‚zt‰gs“fn Riemann
‚zrŠuŒ‹‡gp†v‚„us†† f : X → Y
wBgdu†rŠgd‡{™„†p…v†vt‰gdy¥t‰… Y
ˆYgp†’t‰…
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Riemann
tfgsygpidpY‚ž”Š™fu{wŠŽdy}wBgsjdrƒnsidgdy C
Ž%t‰…Šq‚rƒ…sidqpy††pndjd…Šq F
cV|}x€id…Šytjƒx€ˆYy¬jŠ‡{gsyg
†pgs‹T…Š™fesy'ˆktvuy qsrƒ‚€™„”ŠgptvuwBx€yV‚zrŠ‚zwŠtdedˆY‚‰uyV”didx€rƒ…Šqsjƒ‚V—stfud…X”ƒgp¡Tjƒ—ƒy qsrƒ‚€™„”ŠgptvuwB—ptfgpt‰gdy†t‰…ŠqˆYsVjdgst‰…Šy$t€y††ijƒ‚€™f—ƒjƒ…Š™f‹Yy††‚ˆ‡qv†pgs™dtfŽsˆ‡‚zy††i‚zrŠ‡ƒt„…ƒq C
uˆY…Šnpt‰gƒuƒjd‚}tfgv†£jƒ…v†ped—dg¨§ƒ—fgsidgs—fŽwBev¡G‚ ˆYqsjdrŠgppYŽdy wŠgƒuƒˆYqv†p‚zwŠtvuwBŽ‚zrŠuŒ‹‡ep†v‚„u{g Riemann
‚‰‡†pgƒuŠgh‚zrŠuŒ‹‡ep†v‚„u{g Riemann
jƒ‡ŒgsygsispY‚ž”Š™vuwBŽsy$wBgdjsrŠnpiƒgsypc‘‚ztfeagprŠ—gsqstde¨§ƒt„…rƒ™dsRt‰…¥wŠ‚‰‹‡esidgdu…%wBgduY…Šu‡‚zrBu{‹Tev†p‚‰u{‚€y Riemann
‚‰‡†pgƒuk—fns…¥gp†p‚
“fes™dtfgptf‚€yiwBgƒu gsqstd—p†v…Šjƒ‚€y£jƒgv¡GgsjƒgptvuwBx€yi…p†ftf—ptfgptf‚€ysci|}…ƒqsided³Yuˆkt‰…p†agdqptfŽ¤Žpt„gv†¥g¤—vuwBŽjƒ…Šq%esrƒ…%Tgkc¶l@tfgv†‚gd™d³YŽ c„c‰c'idxzrŠ…Šqsjƒ‚ tdgp†£ˆYqdˆY³kxztvuˆYg£tdgdy¦…Šjded—fgsy$t‰y†† deck transformations
‚ž†p—ƒytgv†pgpiƒqptvu
wB…Šn¤wBgsidnsjƒjƒgpt‰…Šy rƒ…Šqm‚zrŠeppY‚€t‰gdu gsrƒ—©xz†vg t‰…ƒrŠ…pid…sp‡uwB—~wBepidqdjdjƒg¥wBgƒuPgp†ftvuˆktf™f—ƒ‹‡yVy
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jƒ‚%tdgp†~…Šjded—fg Galois
tfgsy‚zrŠxzwŠt‰gsˆ‡gsy‚t€y††~gp†vtf‡ˆkt„…Šu³By††mjd‚‰™d—Šjƒ…Š™d‹‡y††©ˆkyVjƒept€y††~wŠgƒurŠgs™f…ŠqsˆTu{esf…Šqsjƒ‚}jƒ‡Œg‚‚€‹Tgs™fjƒ…ppYŽ%gsqstdŽdy$tfgsyXˆYqd³kxztvuˆ‡gsytˆkt‰…¥gp†vtf‡ˆktf™f…Š‹‡…arŠ™f—s”ƒidgdjdg£tdgdy¡G‚zyV™v‡{gsy Galois
c'‰VrB‡ˆYgsywBep¡T‚‚zrBu{‹‡ep†p‚‰u{g Riemann
jdrƒ…Š™f‚‰‡Y†pgt‰gs“vu†p…Šjƒgv¡T‚„‡‡jƒ‚t”ƒedˆYgt‰…p†%wBgp¡T…piŠuwB—¥wBgpiƒqprƒtfu{wŠ—¥tdgdyi³BsV™f…!XhuTrŠu¡Ggv†p…B‡‡wŠgp¡T…siƒuwB…B‡kwBgsidqprƒtvuwB…Š‡‡³BsV™f…Bu§GˆYgv†gsrdiƒe¤ˆYqp†p‚zwŠtfuwBx€y‚‚zrBu{‹‡ep†p‚‰u{‚€y‚‚‰‡†pgduPjd—p†v…‘td™f‚‰uy'
unifornization
¡T‚€sV™dgdjdg p‡u{g~‚zrBu{‹Te
†p‚‰u{‚‰y Riemann
c² †vyV™v‡{f…v†vt‰gsy£t‰…Šqsy‚gsqpt„…ƒjƒ…Š™f‹TuˆYjƒ…Šnsyt€y††¥wBgp¡T…piŠuwŠs††%wBgsidqprƒtvuwŠs††³BsV™dy††jsrŠ…Š™f…ƒndjd‚$†pg%”dpYesid…Šqsjƒ‚$ˆYqdjsrŠx€™fgsˆ‡jdg%p‡uŒg‚tfgv†%‚zrŠuŒ‹‡ep†v‚„u{g‚jƒgsyscu¦‚‰™‰†pedjd‚Xxztfˆ‡ukˆYt‰…tf™v‡t„…awŠ‚‰‹‡esidgdu…%—ƒrƒ…Šqh¡GgagsˆY³Y…pidgp¡T…Šnsjƒ‚}jƒ‚¦tvuy Fuchsian
…Šjde
—f‚€y rŠ™dsRt‰…ŠqX‚„‡{—f…ƒqdypc‰@‡†pgduƒ—vu{gswŠ™vutfx€y@qsrƒ…Š…Šjƒes—f‚€y'tdgdy SL2(R)
§drƒ…Šq—pt‰gp†X—f™fesˆ‡…ƒqp†XˆYt‰…rŠev†vy gv†p…BuwŠtd—‘jŠupYgs—vuwB—‘gdjƒuŒ‚zrŠ‡{rƒ‚€—f… H∗ rƒ…Šq~t‰…Šqmx€³k…Šqsjƒ‚‚rƒ™f…ŠˆB¡Gx€ˆY‚‰u3t‰g cusps
tdgdy…Šjƒes—fgsy
§ptf—ptf‚ …i³BsV™f…Šy}rƒgsiƒ‡wB…rŠ…ŠqtrŠ™f…ƒwŠnsrdtf‚‰us‚‰‡†pgdudˆYqdjsrŠgppYŽdypc¦qptf—ijƒgsy}…Š—fgppY‚„‡sˆ‡‚ˆ‡qsjdrƒgspY‚‰‡y ‚zrŠuŒ‹‡ep†v‚„u{‚€y Riemann
c†t™f³YuwBe”diƒxzrŠ…ƒqdjd‚¶tfuy}t„…drŠ…piƒ…pp‡u{wŠx‰y@…Šjƒes—f‚€y}wBgduƒˆ‡q
†p‚€³k‚„‡y}—f™fesˆ‡‚‰uy$jŠ‡{gsy t‰…ƒrŠ…piƒ…pp‡uwBŽdy@…Šjƒes—fgsy rŠev†vy”ˆY‚}xž†pgp†Xt‰…ƒrŠ…piƒ…pp‡uwB—h³BsV™f…Tc@hX™v‡{f…Šq
jƒ‚ t‰…ƒrƒ…sid…sp‡uwB…Šnsy ³BsV™f…ƒqdy%rƒgsiƒ‡wBg¥rŠ…Šq¤rŠ™f…Šx€™f³k…v†vt‰gƒu6gprŠ—©—f™fesˆY‚„uy%qprŠ…Š…Šjded—„y††¥tfgsyt‰…ƒrŠ…piƒ…pp‡uwBŽdy'…Šjƒes—fgsy ˆYqdjsrŠgpp‡‚‰‡y §d—vu{gswB™futfx€y cehep†p…ƒqdjd‚ t„gs“vu†p—ŠjdgdˆYght€y††ip‡™dgdjdjŠuwŠs††wŠidgdˆYjƒgptvuwŠs††Ejƒ‚zt‰gsˆ‡³kgsjƒgptvuˆYjds†† wŠgƒu@t€y†† rBu†pepwŠy†† rƒ…Šq gsqst‰…B‡¶gp†vtfurŠ™f…ŠˆkyRrŠ‚€ns…Šqv† ˆ‡‚%qsrƒ‚‰™‰”Š…piŠuwB…ƒndy §GrŠgs™fgv”Š…piƒu{wŠ…ŠndyiwBgdu3‚zisiƒ‚‰urƒtfu{wŠ…ŠndyiwBgdu3jd‚€id‚ztfesjƒ‚£t„g~ˆYt‰gv¡G‚€™femt‰…Šqsyˆ‡gsjƒ‚‰‡{git„g‚…ƒrŠ…Š‡Œg !fwŠidgd™f…v†p…Šjd…Šnv†Xt„…—v†p…Šjdg‚t‰…Šq‚jƒ‚zt‰gdˆY³kgdjdgstfuˆ‡jd…ŠnhrŠ…Šqit‰g%ˆkt‰gp¡T‚€™f…
rŠ…Bu{‚‰‡c|}g¦rŠgs™fgv”Š…piŠuwBe}t‰gX…v†p…Šjdedd…Šqdjd‚ cusps
wŠgƒus‚€rŠuwB‚ž†vtf™ds††v…Šqdjd‚Rt„…i‚ž†p—vu{gs‹Tx€™f…v†tjƒgsyˆ‡‚¦—vu{gpwB™vutdx‰y$qprŠ…Š…ƒjƒes—f‚€y$tdgdy SL2(R)
wŠgƒuku{—vu{gƒ‡tf‚€™fgiˆYtdgp† Γ(1) = SL2(Z)
c@hX™v‡{f…Šqsjƒ‚jŠ‡{ght‰…ƒrŠ…piƒ…pp‡‡{gˆYt‰…v† H∗
= H ∪ {cusps}
tdgdy Γ(1)
wBgduŠjƒ‚ziƒ‚ztdedjd‚ t„…‚rŠgpiŠ‡wB… H∗
/Γ(1)
§
rŠ…Šq¦‚‰‡†pgƒupg¦ˆ‡qsjdrƒgspYx€yscP‰@‡{jƒgsˆYtd‚¶rŠuŒg¦ˆY‚V¡Tx€ˆ‡g$†pgtt‰…Šqt—dsVˆ‡…ƒqdjd‚¶tdgp†¦jƒupYgd—vuwBŽ$—f…ƒjƒŽtrƒ…Šq¡Ggt‰…hwBev†p‚‰udjƒ‡ŒgXˆ‡qsjdrƒgspYŽ‚zrBu{‹‡ep†p‚‰u{g Riemann
wBgdudˆYqv†p‚zrƒsVy}jŠ‡{ggpisp‡‚ž”ƒ™vuwBŽ$wBgdjsrŠnpiƒgkc‘x‰ˆYgagprŠ—tfgv† modular
…Šjƒes—fg%pB†vyV™f‡Œd…Šqdjd‚Xtfgv†£¡G‚€jƒ‚ziƒusV—fg%rŠ‚€™vu…d³kŽ‚tdgdy §kgprŠ—¥—ƒrŠ…ƒqjdrƒ…Š™f…Šnsjƒ‚}†pgax€³k…Šqsjƒ‚¦jŠ‡{g%wBgpiƒŽ£pY‚€yVjd‚ztf™vuwBŽi¡Gx€gsˆ‡g%tdgdy modular
wBgsjdrƒnsidgdy X(1)
g…ƒrŠ…Š‡Œg ‚‰‡†vgƒu6gmgpidpY‚ž”Š™fu{wŠŽ¥wŠgdjsrŠnpiƒg¥tfgsy%rŠgs™fgprŠev†vy Riemann
‚zrBu{‹Tev†p‚‰u{gdy
c£u¦‚€™„†pesjƒ‚ˆYtfuy modular
wBgƒu weakly modular
ˆYqv†pgd™„tfŽsˆ‡‚‰uy §ŠrŠ…ƒq£¡Tg%jƒgsyt”ƒ…Šgv¡GŽsˆ‡…ƒqp††vg%—f…Šnsjƒ‚—stfu†g j
Xgp†pgpisiƒ…B‡yRt‰…Šy‚ˆ‡qv†pes™dtfgsˆ‡g©‚‰‡†pgdu†jŠ‡{g modular
ˆYqp†pes™dtdgdˆYgmrŠ…Šq©‚zrƒespY‚‰u3xž†pgv†8gv†pgpiƒqptvuwB—
uˆY…Šjƒ…ƒ™f‹GuˆYjƒ—©t‰…Šq X(1)
jƒ‚£t„… P1
(C)
c¥ehgst‰gpiƒŽppY…Šqsjƒ‚iˆYt‰… uniforniza-
tion
¡T‚zsV™fgdjdg‚p‡u{g‚‚zisiƒ‚‰urƒtfuwBx€y wBgdjsrŠnpiƒ‚€y §ƒjd‚¦t‰……ƒrƒ…B‡…‚jdrŠ…ƒ™f…Šnsjƒ‚ †pg‚t‰gsqstf‡ˆY…Šqdjd‚ t‰…p†³BsV™f…’t€y††~wŠiƒesˆY‚€y††~uˆY…Š—fqv†pgsjŠ‡{gsy%t€y††m‚zisiƒ‚‰urƒtfu{wƒs††¥wBgdjsrŠnpidy†† 8uˆY…Š—fnv†pgdjd‚€y%wBgsjdrŠn
iƒ‚€ygv†vtvuˆYt‰…Bu³k…Šnv† ˆ‡‚awŠgdjsrŠnpiƒ‚€yajd‚atfgv†©‡{—vu{g j
tgp†vgsisiƒ…Š‡{yRt‰…‘Ž‘ˆ‡‚…ƒjƒ—d¡T‚zt„g lattices
jƒx€ˆky®tdgdy ℘
ˆYqv†ped™„tfgsˆ‡gsy
jƒ‚£t„…v† H/SL2(Z)
c|}xzid…Šy%ˆYqsjdrŠ‚€™fgd‡†p…Šqsjƒ‚i—ptvu3—d‚z† ‚‰‡†pgdu

¡ £ ¥§¡ ¨© ¡ · xi
—fqp†vgstd—p†i†pgaqprŠes™f³k‚‰uYˆYqsjdrŠgppY…ƒrŠ…Š‡gdˆYght„…ƒq%³BsV™f…Šqat€y††‚‚zidid‚‰u{rdtvuwŠs††hwŠgdjsrŠnpidy††£³ByV™v‡y†pgˆ‡qsjdrƒ‚€™vuiƒev”Š…ƒqdjd‚}ˆ‡‚gsqptf—v†%jƒ‡{gu{—vu{—Šjƒ…Š™d‹Tg%wBgdu‡ˆYqv†p‚zrƒsVyjdga‚zidid‚‰u{rdtvuwBŽiwBgsjdrŠnpidgYc¡
‰R†pgijƒ‚zpYesid…‚‚€qs³kgd™vuˆYt„s”ˆYt‰…Šqsy}wBqs™v‡…Šqsy Xcd‘‚€t‰gs‹‡tdˆ‡ŽhwBgduƒicŠ©rŠ‚zidgsp‡u{ep†d†pgYc†xXgŽp¡T‚ziƒg’tfxziƒ…ƒy †pg ‚€qs³Ygs™vuˆktfŽsˆky t‰…v† —dedˆkwBgpiƒ—”jd…Šq ¦™vuˆktf‚‰‡{—fg©§'rŠ…ƒq ˆktfev¡GgpwB‚ rƒep†ft„g—v‡rƒidghjƒ…ŠqiˆY‚}gdqptfx€y'tfuy$—fgdjƒu…Šqs™dp‡uwBx€y'wBgdudrŠgs™fepididgsidg¦—fndˆkwB…piƒ‚€y}ˆktvupYjdx‰y rƒ…ŠqrŠx€™fgsˆYgwBgstdehtfgv†iˆ‡qppkpY™fgd‹‡ŽYc|}…ajƒ‚€™fepwkuƒwBgduŠgigspYeprŠghp‡u{gitdgp†—f…Šqpiƒ‚‰u{et‰…Šq¡Gg£jƒ…ƒq£gs‹TŽsˆY…Šqp†tvuywBgpidnstd‚‰™d‚‰y¦gv†pgdjp†pŽsˆ‡‚‰uy §B‚ž†vtfqprƒsVˆY‚‰uypc
ltc ehgs™fgv†suwB…sid—ƒrƒ…Šqsid…Šy § l'esjƒ…Šyw dµkc

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Galois
l@tf—s³Y…ƒyhjƒgsyh‚„‡†pgduk†vg‹‡tfesˆY…Šqdjd‚Xˆktfgv†aˆYqsˆ‡³kxztvuˆYgat€y††%wŠgsidqsrdtvuwŠs††£gprŠ‚‰uwB…v†s‡ˆY‚€y††‚jd‚t„…¡T‚‰jd‚ziŠusV—f‚€yt¡T‚zsV™fgdjdg%tfgsyt¡T‚zyV™v‡{gdy Galois
cl@t„…‘rŠ™„sRt„…’jƒx€™f…ƒy¡Tg©—f…Šnsjƒ‚£wBeprŠ…Bu{g ”ƒgdˆ‡uwBe~ˆYgsjƒ‚‰‡{gmtfgsy%¡G‚zyV™v‡{gsyat‰y†† wBgpiƒqprƒtfuwŠs††gsrƒ‚„uwB…v†s‡ˆY‚zy††Bc¶l'gv†i”Šesˆ‡g%x‰³k…Šqsjƒ‚$rŠes™f‚‰uYtdgp†‚‚€™dpYgsˆT‡{g%t‰…Šq65w87 cl@t„…‚—f‚€nstd‚€™f…‚jdx€™f…Šy t‰…ŠqrŠgs™f—v†vt‰…Šy wB‚€‹Tgpidgƒ‡…ƒq¦¡Tg£—d…Šndjd‚ ˆYqp†p…drƒtvuwBe©§pwBeprŠ…Bu{gˆkt‰…Bu³k‚‰‡Œggsrƒ—’tdgp†¤¡G‚zyV™v‡{g Galois
§PpTu{g~†pg©‹Ytfesˆ‡…ƒqdjd‚%ˆkt„…‘¡G‚€jd‚€iƒusV—f‚€y‚¡T‚€sV™dgdjdgmtfgsy%¡G‚zyV™v‡{gsy
Galois
c ¡
‰VrŠ‚‰ut‰g ¡Tg’‚zrŠu³k‚„u{™fŽsˆY…Šqdjd‚£†pg©gv†pgs—f‚‰‡{“f…Šqsjƒ‚%t‰…Šqsy—f‚€ˆ‡jd…Šnsygp†pesjƒ‚€ˆYg©ˆYt‰…Šqsy—fnd…¤gsqst‰…Šnsytjdgp¡TgsjƒgptvuwB…ŠnsytwŠides—f…Šqsysc
9A@B9 CEDGFGHPIRQTSVUXW`YbacIedfSVUXgThpirqXdfSsY
¡
htrŠ…Šq$—f‚ž† gp†pgs‹Tx€™f…v†vt‰gdudˆ‡qppkwB‚zwB™vu{jƒxž†pg}ˆkt‰…Bu³k‚‰‡{g pTu{g t‰…ŠqsyR³BsV™f…Šqsy¶rƒ…Šq'¡Gg}rƒ™fgspYjdgstd‚‰q
t„…ƒndjd‚ˆ‡‚Xgdqptf—t‰…¤wŠ‚‰‹‡esidgdu… §ƒ¡Tgqsrƒ…d¡Tx€t‰…Šqsjƒ‚¦—ptvuT‚‰‡†pgduT—f™f…Šjd…ŠˆYqp†p‚zwŠtfuwB…B‡YwŠgƒukt„…drBuwBe—f™f…Šjd…Šˆ‡qv†p‚zwŠtfu{wŠ…B‡cVxXghgs™f³Y‡ˆ‡…ƒqdjd‚ qsrƒ‚ž†f¡GqsjŠ‡{f…v†vt‰gsy xž†pg¡G‚zsV™fgsjƒgigprŠ—itvuy wBgpidqsrdtvuwBx€ygsrƒ‚„uwB…v†s‡ˆY‚‰uy §ŠwBgv¡TsVyXwŠgƒuYxz†vgaedjd‚€ˆ‡…rŠ—ƒ™vuˆYjƒgagsqst‰…ŠnXt
WEF‡a`vu ¨ (xw (`wPy€ p : (Y, y0) → (X, x0) ‚Bƒ€B„†…ˆ‡‰„‘’ˆ“•”–…—‡™˜d„•“ ‚ …—‡ê8€f…gihj‡‰„k…‰l Y m €n„‰o
pVq l‘rxlfgv’k€ ‚ ‡ˆ”–…s‡‰efout0v q lfonw3”xe–” ‚ ’ˆ“fy qt ‚ …zr`…{„ ‚ “ ƒ „•“ ‚ …s‡ê8€8…|gzh
φ : π1(X, x0) → p−1
(x0).
y€Xl Y ‚Bƒ€B„†…i„•“}zyXgv’k€ ‚ ‡ˆ”–…s‡‰efod”xe–” ‚ h φ ‚Bƒ€B„†…~8€~j‡‰„k… ‚ “ ƒƒ‚
’‚ p∗
‚ž†f†p…Š…Šnsjƒ‚¶t‰…p†‚€rƒgspY—Šjd‚ž†p…ˆ„§ƒgprŠ—htdgp†XwBgsidqprƒtvuwBŽ¦gprŠ‚‰uwB—p†puˆ‡gX…Šjƒ…Šjd…Š™f‹Tuˆ‡jd—Tc
«†…R`† ¨ (`w„‡w
i|ˆ p∗ : π1(Y, y0) → π1(X, x0) ‚Bƒ€B„†… m €n„‰oerxl8€xlBr‰l q‹Š …|g‹r3efo ‚

iiŒ “fy qt ‚ …zr ƒ„Ž~f€V~‡‰„†… ‚ “ ƒ „“ ‚ …s‡‰ef€8…|gzh
Φ : π1(X, x0)/H → p−1
(x0)
e•“l–’ H = p∗(π1 Y, y0)
‡‰„†… π1(X, x0)/H
p h8‹v0€ ‚ …x”8h‘gv’‘kil’˜ p ‚n“ …|v%€”gv’€
r‡“nief‡†•%€”Bl–’ H
gi”xh‹€ π1(X, x0).
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v · ¢¡¤£¦¥¨§¦¥©¢¥¡ ¥©¢ ¡ ¥© !¥! ¡#£¦ Galois
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π1(Y, y0)/Kerp∗
∼= Imp∗
wŠgƒu es™fg π1(Y, y0) ∼= p∗(π1 Y, y0) = H.
}†¤g H
‚„‡†pgduwBgv†p…p†pu{wŠŽXqprŠ…Š…Šjded—dg¦tdgdy π1(X, x0)
tf—ptf‚@g p
…v†p…Šjdedf‚zt‰gdupwBgv†p…v†suwBŽXŽ Galois
wBgpiƒqprƒtfu
wBŽ¤gsrƒ‚„uwB—v†suˆYgawBgdu …m³BsV™f…Šy Y
wBgv†p…v†suwB—Šy£Ž Galois
wBgpidqsrdtvuwB—Šyh³BsV™f…Šypchu¦™fxzrŠ‚‰uT†pgrŠgs™fgptfgs™fŽdˆY…Šqsjƒ‚i—ptvu6gmˆ‡qv†f¡TŽswŠg tfgsy%wBgv†p…v†suwB—ptfgpt„gsy£‚z†v—Šy%wBgpiƒqprƒtfuwB…Šn¥³BsV™f…Šq©§3‚‰‡†pgdugp†v‚‰“ded™dtdgstdg¥gsrƒ—~tfgv†¤‚zrBuiƒ…ppYŽt„…ƒq y ∈ p−1
(x0)
ci¦qst‰…B‡ …Bu6³BsV™f…Bu§G¡Gg jƒgsy‚gprŠgsˆY³Y…
iƒŽsˆY…Šqp†‚u{—vu{gd‡tf‚€™fg£rŠgs™fgpwBest€y}c
$ ¨'¨ (`w¦% $ ¨@¨ 3I‡…R`('X#auu†A”ªG``… ¨ #‚I0) w
12 gz”}• p : Y → X
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y65
‚ pVq eBrxlfo f : I →
X
r ‚ „q tG˜‘”BlXgzhˆr ‚Bƒl x0 mt ‚ …rxlf€n„ p …s‡™˜6„†€fe q 5f•%gzh ˜f
gi”Bl8€ Y
w‡r ‚ „q tG˜‘”Blgzh™r ‚‘ƒl
y0 ‚7
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iƒŽsjƒjdgst‰…Šy
$ ¨'¨ (`w¯ w 12 gi”¸• p : Y → X
‡‰„‘’ˆ“•”–…—‡™˜ „•“ ‚ …s‡ê8€8…|gzh6‡‰„k… Z
gv’k€ ‚ ‡™”‘…s‡ê8o”Bl•“lsvl–€
’‹…s‡‰efoAt%v q lfo ‚ 12 gz”}• ‚ “ ƒgih™o ˜f1, ˜f2 : Z → Y
gv’†€ ‚t ‚‘ƒo „“ ‚ …—‡ˆl8€ ƒg ‚ …|oj” m ”Bl8…‚ oev%gz” ‚98
p ◦ ˜f1 = p ◦ ˜f2 ‚ yX€ ˜f1(z) = ˜f2(z)
’‹…{„ m €B„Žgzhˆr ‚Bƒl z ∈ Z
w‡”–ex” ‚ ˜f1 = ˜f2 ‚
y“•e p ‚ …“ h 8
t™„wB‚‰‡‡†pg¥—f‚‰‡Œ“d…Šqdjd‚h—stfuTt‰…~ˆYnp†p…pid… ˆkt‰… Z
§Yp‡uŒgt‰…~…drŠ…B‡…¤…BuGgprŠ‚‰uwB…v†s‡
ˆ‡‚‰uyhˆYqsjƒ‹‡y††v…Šnp†a‚‰‡†vgƒuGgv†p…BuwŠtd—¥wŠgƒu‡t‰… ˆ‡qsjdrdiƒŽs™dyVjƒe‚t„…ƒq¨§‡ˆYt‰…¤…ƒrŠ…Š‡… …ŠuTgsrƒ‚‰u{wŠ…p†s‡ˆ‡‚‰uy—vu{gd‹Yy††p…Šnv†i‚‰‡†pgƒuB‚zrB‡ˆ‡gsy$gp†p…Šu{wƒtf—Tc’‚¦gsqstd—p†ht‰…v†htf™d—ƒrŠ…%t‰…aˆYnv†p…sid…a—ƒrƒ…Šq¡Gg£ˆYqdjd‹‡y
†p…Šnv† §G¡Tg~‚‰‡†pgƒuGt‰… ∅
UrŠ™feppYjƒg~ept‰…ƒrŠ…~gsrƒ—~tfgv†¥qprŠ—d¡T‚€ˆ‡g
Ž¤…sid—ƒwŠidgs™f… t‰… Z
c ¡
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w ∈ {z ∈ Z : ˜f1(z) = ˜f2(z)}
c¦‰VrBuidx€pky N ⊂ X
§Tgp†p…Šu{wƒtfŽpY‚„ut„…v†su{et‰…Šq p ◦ ˜fi(w)
§
jƒ‚ i = 1, 2
g…ƒrƒ…B‡{ga‚‰‡†pgƒu‡…ŠjƒgpiƒeawŠgsidqdjdjƒxž†pg‚gsrƒ—¤tdgp† p
c'utgd‡{™„†vy p−1
(N)
†pg¥‚‰‡†vgƒuTg“fxž†pg%xž†vyVˆYggv†p…BuwŠtds†† Na
tfxzt‰…Bu{g‚sVˆktf‚tp‡u{g‚wŠep¡T‚ a
g p|Na : Na → N
†vg%‚‰‡†pgƒuYxž†pgdy…Šjƒ…Šu…Šjƒ…ƒ™f‹GuˆYjƒ—ƒysct¦rƒ—~tfgv†¥ˆYqp†vx‰³k‚‰u{gt‰y††ˆ‡qv†pgs™dtfŽsˆ‡‚zy†† ˜f1, ˜f2
§T¡Tg¤rƒ™fxzrŠ‚‰uGg ‚‰uwB—v†pg‚ž†p—ŠyˆYqp†p‚zwŠtfuwB…Šn©ˆYqp†p—pid…Šq †pg©‚‰‡†pgduˆ‡qv†p‚zwŠtfu{wŠ—EˆYnp†v…sid…’wŠgƒuPˆ‡qv†p‚zrƒsVya¡TgmrŠ™fxzrŠ‚‰uP†vggsrƒ‚‰u{wŠ…p†s‡{f…ƒqp†‚jŠ‡{gpY‚‰ut‰…p†su{e V
t‰…Šq w
ˆYt‰…~‡Œ—fu… Na
c}ehgv¡TsVyi—ŠjsyVy p ◦ ˜f1 = p ◦ ˜f2
§‡…Bu
˜f1,
wŠgƒu ˜f2
§p¡TgXrƒ™fxzrŠ‚‰uv†vgXˆYqdjd‹‡y††p…Šnv†tˆYt‰… V
c ¡
‰Vtfˆ‡usp‡u{gttdqd³kgd‡… w ∈ { ˜f1(z) = ˜f2(z)}
qsrƒed™d³Y‚‰ugv†p…BuwŠtdŽmpY‚„ut„…v†su{emt‰…Šq V
§†jƒ‚ V ⊂ {z ∈ Z : ˜f1(z) = ˜f2(z)}
wŠgƒu†xztfˆ‡ut‰…
{z ∈ Z : ˜f1(z) = ˜f2(z)}
‚‰‡†vgƒuYxz†vgagp†v…BuwŠtf—¥ˆYnp†p…pid…Tc¡
hXjƒ…ŠuŒg¤gv† t‰… w ∈ {z ∈ Z : ˜f1(z) = ˜f2(z)}
§6…Bu ˜fi
¡Tg rƒ™fxzrŠ‚‰uT†pg~gprŠ‚‰uwB…v†s‡{f…Šqv†¥jŠ‡{ggp†v…BuwŠtfŽhpY‚‰ut‰…p†s‡{gt‰…Šq©§ V
ˆ‡‚$—vu{gs‹T…Š™d‚€tfuwBehwBgƒuŠˆYqv†p‚zrƒsVy¦“fxž†pg Na
c ¡
‰VtdˆTuƒ¡Tg£rƒ™fxzrŠ‚‰ud†pg—vu{gd‹Yy††p…Šnv†‚ˆYt‰… V
wBgdu‡es™fg%t‰… {z ∈ Z : ˜f1(z) = ˜f2(z)}
‚‰‡†pgdu‡gv†p…BuwŠtd—Tc♦
WEF‡%ù ¨ (`w ° w 12 gi”¸• p : (Y, y) → (X, x)
€B„ ‚Bƒ€B„†…G‡‰„‘’ˆ“•”–…—‡™˜ „•“ ‚ …s‡ê8€8…|gzh r ‚
p(y) = x
‡k„k… f, g
pVq e‘rxl8…xgi”Bl8€ X
r ‚ „q tG˜ ”Bl x
‡‰„†…x” m vl8oX”Bl x1 ‚ yjoŽ’ˆ“•l@5
m g™l–’xr ‚
‚ “ ƒgzh™oe–”–… ˜f, ˜g
l8…3„†€” ƒgi”Bl8…t ‚ oX„†€xl q 5v%g ‚ …|o””Bl–’konwGl8…‡l•“•l ƒl8… ‚Bƒ€B„†… pVq e‘rxl8…fgz”Blf€ Y
r ‚„qt G˜‘”Bl y‚ y€ f p g
w‡”xe–” ‚ l8… ˜f, ˜g mtAl–’†€”‘l ƒp …|l” ‚ z…s‡êjgzh™r ‚‘ƒlBA¨“•l‘’”„†€˜™‡ ‚ …gi”Bl8€
Y C
‡‰„k… ‚Bƒ€B„†… pVq lBr‰l‘€VlBr‰l–”Bl•“†…—‡ˆl ƒƒ‚D
¡
‰¶ˆkt‰y y1, y2 ∈ Y : p(y1) = p(y2) = x0
cVu$sVyh…Bu‡‚‰uwB—v†p‚€yXt€y††ajd…p†p…ƒjƒ…Š™f‹TuˆYjds††
p∗ : π1(Y, y0) → π1(X, x0)
wBgdu p∗ : π1(Y, y1) → π1(X, x0)
¡Gg©jsrŠ…Š™d…ŠndˆYgv†¤†vgmˆ‡qppkwB™vu¡T…Šnv†FE‰VrBuiƒxzpky [γ]
jƒ‡Œg wŠiƒesˆYgm—f™f—Šjsy††mˆkt‰…p† Y
jƒ‚%gd™f³kŽt‰…mˆ‡gsjƒ‚‰‡… y0
wBgduGtfxziƒ…ƒyit‰… y1
c£hX™v‡{dy®uˆY…Šjƒ…ƒ™f‹GuˆYjƒ—6G
u : π1(Y, y0) → π1(Y, y1)
jd‚
H ¥‘¦‘§¨œ–ŸT©ª}«‰¦PI‘µxµ–™Tº8±|¹PQ¬Ÿ¨¦– ž}™AªPR8±SUT ˜™%šWVn¤A™nœ–¶–ž}ŸV˜¡XVT¥‘¦‘§¨œ–ŸT©ª}«6Y†Ÿs£´XV‘µ–™Aº8±|¹PQ¬Ÿ¨¦– ž}™A8¹B±` ™n¦B¦baPçŸV˜†šWVn¤%¥9aB¬@VA®–¦‘ŸV˜¬fš}£™¤Aµ–}¤B‘œ–™nšn˜£‹¤0šVx¢A°8£z´¸x¢ Y
Q8¥‘¦‘§¨œ–ŸT©ª}«ˆx´}˜¬µ–¶–›38±ªdQ•¬Ÿ¨¦– ž¸™R¹8±

¡ ‘ !¡9£¦¥¨§¦¥©!#¥ ¡¦¥©¢# ¡ ¥ ·
˜
tfnprŠ…! u(a) = [¯γ] ∗ [a] ∗ [γ]
wŠgƒukrŠgd‡Œ™‰†vy¬t‰…rŠgs™fgpwBept‰y jd‚€t‰gv¡G‚ztfu{wŠ——vu{espY™fgsjƒjdg ¢
π1(Y, y0)
p∗
−→ π1(X, x0)
↓ u ↓ v
π1(Y, y1)
p∗
−→ π1(X, x0)
•–ˆ‡³kns‚„uR—ptvu v(b) = (p∗[γ])−1
∗ [b] ∗ (p∗[γ])
c ¡
hXjdyVy p∗[γ] = p ◦ [γ]
¡TgE‚‰‡†pgdu†xž†pgwŠiƒ‚‰uˆktf—‘jd…p†p…drŠeptvu6jƒ‚‚”ƒedˆYg t‰… x0
§PwBgƒuPˆ‡qv†p‚zrƒsVy£¡Gg©gv†pŽpwB‚‰uPˆktfgv† π1(X, x0)
c ¡
‰VtdˆTu…BuG‚‰u{wŠ—p†p‚€yt€y†† π1(Y, y0)
wŠgƒu π1(Y, y1)
jƒx€ˆky tfgsy p∗
‚‰‡†pgƒuGˆYqsfqspY‚‰‡yqprŠ…ƒ…Šjƒes—f‚€ytfgsy
π1(X, x0)
c‘rƒ…Š™f‚‰‡'wBev¡G‚©qprŠ…ƒ…Šjƒes—fg ˆktfgv† wŠidesˆ‡g”ˆYqddqsp‡‡{gdymtdgdy‘qprŠ…Š…ƒjƒes—fgdy p∗(π1(Y, y0))
†vgrŠ™f…ƒwŠn%‡‚„u ˆYgv† tfgv†”‚‰uwB—v†pg p∗(π1(Y, y1))
jƒ‚mtdgp† ‚€rŠuiƒ…ppYŽ ‚ž†p—Šy©wBgptfepididgsid…Šq y1 ∈
p−1
(x0)
E%• gprŠev†vtdgdˆYg ‚‰‡†pgdu¤£¦gduc£ehev¡G‚£qprŠ…Š…Šjded—dg¤ˆY‚%gsqptfŽv†tfgv†¤wŠiƒesˆYg ˆYqddqsp‡‡{gdyx‰³k‚‰upjƒ…Š™f‹‡Ž [a−1
]∗[p∗(π1(Y, y0))]∗[a]
§pp‡u{g [a] ∈ π1(X, x0)
c†}† f : I → X
wŠid‚‰uˆYtd—jƒ…v†p…ƒrŠeptvuBwBgduYgv†vtfu{rƒ™f—ŠˆkyRrŠ…Šy$t‰…Šq a
§BgprŠ—at‰…aiƒŽsjƒjdg¤bsc±‚qsrƒed™d³Y‚‰uBjƒ…v†pgs—vuwBŽ£gp†p—ƒ™„¡TyVˆYgˆ‡‚—f™d—Šjƒ… g : I → Y
jƒ‚gs™f³kŽ~t‰… y0
c ¡
‰¶ˆkt‰y y1
†pg©‚‰‡†pgduPt‰…‘tfxziƒ…ƒyat„…ƒqYc¤|}—ptf‚‚¡Tgx‰³k…Šqsjƒ‚t—stfu
p∗(π1(Y, y1)) = [a]−1
∗ [p∗(π1(Y, y0))] ∗ [a].
l'qp†p… %G‡{f…v†vt‰gdyt¡Tgx€³k…Šqdjd‚¦t‰…¥rŠgs™fgpwBest€yDgprŠ…ptfxzid‚‰ˆYjƒg
WEF‡a`vu ¨ (`w¦¥fw 12 gz”¸• p : (Y, y) → (X, x)
‡k„–‹’ˆ“†”‘…s‡™˜ „•“ ‚ …—‡ˆe8€f…gih ‚¨§
…‡’ˆ“lflBr‡y p ‚ o
p∗(π1(Y, y))
w‰’‹…{„ y ∈ p−1
(x)
„•“l–” ‚ il©†€P„k‡ q …iv%oAr ƒ„j‡xzy‰gzhXgv’¡‘’’ ƒ„koA„•“e ’ˆ“lflBr‡y€
p ‚ o ”Bl–’ π1(X, x)‚
¡
‰¶ˆYt€y p : (Y, y) → (X, x)
wBgpidqsrdtvuwBŽhgprŠ‚‰uwB—p†puˆ‡g©§Šjd‚ y ∈ Y
wBgdu ϕ : (Y , y ) →
(X, x)
†pg¥‚‰‡†vgƒuTjƒ‡{g¤ˆYqv†p‚€³YŽsyigsrƒ‚‰u{wŠ—p†suˆ‡gkc u¦—std‚iqsrƒed™f³k‚‰u ˜ϕ : (Y , y ) → (Y, y)
§Txztfˆ‡usVˆYtd‚ag ˜ϕ
†pg©gprŠ…ptf‚ziƒ‚‰‡ gv†p—Š™„¡‡yVˆ‡gmtdgdy ϕ
E¥|}—std‚%gpwB™vu”dsVy%t‰…‘rƒgd™dgswBept€y —vu{espY™fgsjƒjdg¡Gg‚‰‡†pgdu‡jƒ‚zt‰gv¡G‚ztvuwB—
(Y , y )
˜ϕ
−→ (Y, y)
ϕ p
(X, x)
y€n„8’‡‰„ ƒ„ g’†€@5‰˜™‡™h 8
}†’g ˜ϕ
qprŠes™f³k‚„u†td—std‚¥¡Tg x€³k…Šqsjƒ‚t‰… rŠgs™fgswŠest€y˜jd‚€t‰gv¡G‚ztfu{wŠ——vu{espY™fgsjƒjdg¨§Brƒ…Šq%rŠ™d…Šx€™f³k‚€t‰gduYgprŠ—t‰…ŠqsyX‚zrŠgppY—Šjƒ‚ž†p…ƒqdyt…Šjƒ…Šjd…Š™f‹Tuˆ‡jd…Šnsy
π1(Y , y )
˜ϕ∗
−→ π1(Y, y)
ϕ∗ p∗
π1(X, x)
¡
hXjdyVy‘g p∗
‚‰‡†pgdu'jƒ…v†p…Šjd…Š™f‹Guˆ‡jd—Šy©wBgƒu π1(Y, y) ∼= p∗ π1(Y, y)
c ¡
‰VtdˆTu g”nprŠgs™f“fg…Šjƒ…Šjd…Š™f‹Tuˆ‡jd…Šn ˜ϕ∗
rƒ…Šq wBev†p‚‰u't‰… —vu{epp‡™dgdjdjƒg”jƒ‚zt‰gv¡G‚ztvuwB—#§}‚‰‡†vgƒu}uˆY…Š—fnv†pgsjƒg”jd‚mtfgv†ˆ‡qv†f¡TŽswBg
ϕ∗ π1(Y , y ) = p∗ ˜ϕ∗π1(Y , y ) ⊂ p∗ π1(Y, y) .
’epiƒuˆYt‰ghrŠgs™fgptfgs™f…Šnsjƒ‚@—stfukg£ˆYqp†f¡TŽpwBghpTu{ghtfgv†inprŠgs™f“fgijŠ‡{gsy}tfxzt‰…Bu{gdy$gv†p—Š™„¡‡yVˆ‡gsy¦—f‚ž†‚„‡†pgdu‡jd—p†v… gv†pgppYwŠgƒ‡{g%gpisiƒe£‚‰‡†vgƒukwBgƒu …s‡‰„•€•˜
WEF‡a`vu ¨ (`w » w 12 gz”}• p : (Y, y) → (X, x)
‡‰„‘’ˆ“•”–…—‡™˜ „“ ‚ …s‡‰ef€8…|gzhkw‡r ‚ y ∈ Y
‡‰„†…
ϕ : (Y , y ) → (X, x)
€B„ ‚Bƒ€B„†…`r ƒ„ g’†€ ‚tG˜‹oj„•“ ‚ …s‡ˆe8€8…|gih ‚ Œ “fy q t ‚ …rxl8€B„ p …s‡™˜‘„†€fe q €
5•%gzh ˜ϕ : (Y , y ) → (Y, y)
w3”xh™o ϕ
„•€‡‰„k…r3e8€xl8€j„†€ ϕ∗ π1(Y , y ) ⊂ p∗ π1(Y, y)
nAŸ¨œ–™ T £µxŸ ¤nx›3™nœ–¶TšWVn¤ p
–µx–µ–x´k˜¬µ–¶x›3ŸV ¤B™‘˜f™}¤BŸs¡XaB´¸šWVBšVx›u™nœ–¶0šWVn¤0Ÿ¨œx˜¦‘ T I0¥9aB¬@V‘›UQ8¥‘¦‘§¨œ–Ÿ0©ª¸«œx™B´XaBž}ŸV˜T µ–™G8±¤Q¬Ÿ¨¦– ž¸™Aª8±

· ¢¡¤£¦¥¨§¦¥©¢¥¡ ¥©¢ ¡ ¥© !¥! ¡#£¦ Galois
y“•e p ‚ …“ h 8
¦™dwB‚‰‡¶†vg —f‚‰‡{“f…Šqsjƒ‚~—ptvu'uˆ‡³kns‚‰u¶t‰…¬gv†vtf‡ˆYtd™f…Š‹‡…!ExXg —f‚‰‡{“f…Šqsjƒ‚ ˆYtdgp†gd™d³YŽ¤—ptvu gp†gdqptfŽ¤qsrƒed™d³Y‚‰u§‡‚„‡†pgdu jƒ…v†pgs—vuwBŽYc ¡
‰¶ˆYt€y y1 ∈ Y
§6—vu{gpiƒxzpky —f™f—ƒjƒ…mg¤gsrƒ—t‰… y
ˆkt‰… y1
c~u¦gƒ‡{™„†fy t‰…p†~—f™f—ƒjƒ… ϕ ◦ α
ˆkt‰…p† X
§3wBgduPt‰…v†mgv†p…Š™‰¡Ts††vy ˆ‡‚—f™d—Šjƒ…‘pˆYt‰…v†³BsV™f… Y
§Tjƒ‚hgs™f³kŽat„… ˆ‡gsjƒ‚‰‡… y
c $†aqsrƒed™f³k‚‰uTjƒ‡Œggv†p—Š™‰¡TyVˆYg ˜ϕ
t‰…Šq ϕ
§Ttd—std‚Xt‰…
˜ϕ(y1)
§Š¡Gg%rƒ™fxzrŠ‚‰uƒ†pg%‚‰‡†pgdu‡‡ˆ‡…jd‚¦t‰…ap b
§k—fgsidgs—fŽ£t‰…atdx€id…Šy$tfgsytp6c ¡
‰Vtfˆ‡u ˜ϕ ◦ α
‚‰‡†pgdujŠ‡{g£gp†p—ƒ™„¡TyVˆYg%t‰…Šq ϕ ◦ α
§kjd‚tgs™f³kŽ£t‰… y
wBgduB…Bukgp†p…ƒ™„¡TsVˆY‚‰uy¦—f™f—Šjsy††%‚‰‡†pgduYjd…p†vgd—vuwBx€y §
gsrƒ—t„…idŽdjdjƒg¤bdcSbdc±' gawBgstdedˆkt‰gdˆYg%gprŠ‚‰uwB…v†s‡{f‚zt‰gƒuBˆkt„…¥ˆY³kŽdjdg~bdcSb
c¡
‰¶ˆkt‰y y1 ∈ Y
§YrŠgd‡{™„†vy α
†pg‚‰‡†pgdu‡…¤—f™f—Šjd…ŠyiˆYt‰…v† Y
gsrƒ—¤t‰… y
ˆYt‰… y1
§‡—ƒrƒyVywŠgƒu
Y
X
ö
ö
á
â
öoá
ãy
y
x
p
1y
~ ä
1yö( )‘
Y‘
1yö( )‘~
‘
‘
2
y‘
öo â
yö( )‘~
2
N
VW
0
U
l'³kŽsjƒg~bsc‘b
rŠ™vu†Bc@u¦gƒ‡{™„†vy¬t‰…p† ϕ ◦ α
ˆkt„…v† X
§kwBgduYt‰…v†gv†p…Š™„¡‡s††vy ˆY‚—f™d—Šjƒ…¥p©ˆkt„…v†³BsV™f… Y
§‡jd‚gd™d³YŽ‚t„…¥ˆYgdjd‚‰‡… y
c¶hX™v‡{dy
˜ϕ(y1) = γ(1).
„ p ‚Bƒ“ • e–”–…Ph ϕ ‚Bƒ€B„†…Ag’†€ ‚tG˜‹o‘„“ ‚ …s‡‰ef€8…|gzh 8
¡
‰¶ˆYt€y N
pY‚‰ut‰…v†su{e t‰…Šq ˜ϕ(y1)
§
¡Gg©”ƒ™dy˜pY‚„ut„…v†su{e W
t‰…Šq y1
tfxzt‰…Bu{gmsVˆYtd‚ ˜ϕ(W) ⊂ N
c ›au{gpiƒxzpky U
pY‚„ut„…v†su{e©t‰…Šq
ϕ(y1)
ct¦qptfŽ‚‰‡†pgƒuG…ƒjƒgpiƒeawBgpidqdjdjƒxž†pgagprŠ—¥tfgv† p
c ¡
‰¶ˆkt‰y V0
t‰…~‹‡nsisid…¤t‰…Šq p−1
(U)
§
jƒ‚ ˜ϕ(y1) ∈ V0
wBgdu p|V0 : V0 → U
†pg¤‚‰‡†vgƒu xž†pgsy‚…Šjd…Bu…Šjd…Š™f‹Tuˆ‡jd—Šypcu¦gƒ‡{™„†p…v†vt‰gsy§Tgp†³k™f‚‰uŒesf‚zt‰gƒu§„jŠuwB™f—ptf‚€™f‚€y†pY‚‰ut‰…p†puŒx€y†jsrŠ…Š™„s~†pg}qprŠ…d¡Tx€ˆYy~—stfu V0 ⊂ N
c3›au{gpiƒxzpky~pY‚‰ut‰…p†puŒe
W
t‰…Šq y1
§Trƒ…Šq¥‚‰‡†vgƒuG—f™d…Šjƒ…ŠˆYqv†p‚zwŠtvuwBŽwBgduTgv†pŽswŠ‚„uGˆkt‰… ϕ−1
(U)
ch¢trƒ…ŠˆYtdgd™f‡Œ„y tdsV™fg—stfu ˜ϕ(W) ⊂ V0
ca²3u{g~pB†vyVˆktf— y2 ∈ W
§3‚zrŠu{idxzpky®—d™f—Šjd… β
ˆYt‰… W
§3jd‚%gs™f³kŽ~t‰… y1wBgdu tdxziƒ…Šyit‰…©ˆ‡gsjƒ‚‰‡… y2
ciu¦gƒ‡{™„†vy tdgp†¤‚‰uwB—v†pg¤t‰…Šq¤—f™f—ƒjƒ…Šq ˆYt‰…v†¤³ksV™d… X
jdx‰ˆky tfgsy
ϕ
wBgduTgv†p…Š™„¡‡s††vy ˆkt‰…p†a³ksV™d… Y
c ¡
‰¶ˆkt‰y δ = p|−1
V0
◦ ϕ ◦ β
†pg¥‚‰‡†pgƒu‡g¤gv†p—Š™‰¡TyVˆYgt„…ƒq—f™f—Šjd…Šq ϕ ◦ β
jƒ‚igs™f³kŽ t‰…mˆYgdjd‚„‡… ˜ϕ(y1)
ci|}—ptf‚ γ(1) = ˜ϕ(y1) = δ(0)
wŠgƒu ˆYqp†p‚zrdsVy…Š™v‡{f‚zt‰gƒuGt‰… γ ∗ δ
§6t„…©…drŠ…B‡…~‚„‡†pgdu6gmgp†v—Š™„¡‡yVˆ‡g¤t„…ƒq ϕ ◦ (α ∗ β)
jƒ‚‚gd™f³kŽ¤t„…©ˆYgsjƒ‚‰‡…
y
c¬¦rƒ—Dt‰…v†’…Š™vuˆYjƒ— ˜ϕ(y2) = (γ ∗ δ)(1) ∈ V0
c ¡
‰Vtfˆ‡u@p‡u{gEtfqs³kgƒ‡… y2 ∈ W
§'x€³By
˜ϕ(y2) ∈ V0 =⇒ ˜ϕ(W) ⊂ V0
wŠgƒu ˜ϕ
‚„‡†pgdu‡ˆYqv†p‚€³YŽsygprŠ‚‰uwB—v†suˆYgYc „ p ‚‘ƒ“ • e–”–…vh ϕ ‚Bƒ€B„†…z‡k„–zyXl q …gr m €h 8
¡
‰¶ˆkt‰y α, β
§Y—f™f—Šjd…BukˆYt‰…v† Y
§kgprŠ—at‰… y
ˆYt‰… y1
wBgdu ϕ ◦ α, ϕ ◦ β
—f™f—ƒjƒ…BuYˆYt‰…v† X
jƒ‚Xgs™f³kŽ%t‰… x
cV}† γ, δ
…Šu‡gv†p…Š™„¡‡sVˆ‡‚‰uytt„…ƒqdyˆYt‰…v† Y
jƒ‚@gd™d³YŽ¦t‰… y
§sgd™„wB‚‰‡f†vg—f‚‰‡{“dy —ptvu γ(1) = δ(1)
c ¡
‰¶ˆYt€y ε
†pgX‚‰‡†pgdusgtgp†p—ƒ™„¡TyVˆYgt‰…Šq ϕ ◦ ¯β
jƒ‚igd™d³YŽ¤t‰… γ(1)
ch|}—ptf‚ht‰… γ ∗ ε
…ƒ™v‡{f‚zt„gduTwBgdu gsrƒ…std‚€id‚‰‡Ggp†v—Š™„¡‡yVˆ‡g¤ˆYt‰…v†
Y
§ƒt„…ƒq£wŠid‚‰uˆkt„…ƒn£jd…p†v…ƒrŠgptvu…ƒn (ϕ ◦ α) ∗ (ϕ ◦ ¯β)¡
ˆYt‰…v†£³ksV™d… X
cR• wŠidedˆYg‚…ƒjƒ…pt„…drB‡{gdy
¢ ˜¬°¤£BŸV˜•¶nš}˜ ϕ∗(α ∗ ¯β) = ϕ ◦ (α ∗ ¯β) = (ϕ ◦ α) ∗ (ϕ ◦ ¯β)
±

¡ ‘ !¡9£¦¥¨§¦¥©!#¥ ¡¦¥©¢# ¡ ¥ · “
Y
X
ö
ö
á
â
öoá
ã
y
y
x
p
1y
~
äY‘
‘
‘
öo â
å
ø
poø
l'³kŽdjdg bsc{w
gdqpt‰…Šnat‰…ŠqwŠiƒ‚‰uˆkt‰…Šnjƒ…v†p…ƒrƒgstfu…Šn‚‰‡†pgduGg ϕ∗([α ∗ ¯β])
§TrŠ…Šq¥gprŠ—¥tfgv†¥qsrƒ—d¡T‚‰ˆYgrŠ™fxzrƒ‚„u†pg gv†pŽpwB‚‰uGˆYtdgp† p∗(π1(Y, y))
c ¡
‰Vtfˆ‡u qsrƒed™f³k‚‰uTwŠid‚‰uˆYtd—~jd…p†p…drŠeptvu ψ ∈ Y
jd‚”Šesˆ‡gt‰…
y
xztfˆ‡uYsVˆktf‚¦
ϕ∗([α ∗ ¯β]) = [p∗(ψ)].
¡
‰Vtfˆ‡uR¡Tg’rƒ™fxzrŠ‚‰u ε(1) = y
cEu¦™fespYjdgstfuVgprŠ—’t‰…E¡G‚zsV™fgsjƒgDbsc{µ §Rgp†©—fnd… —f™f—Šjd…BuRˆYt‰…v†
X
‚„‡†pgdu—f™d…Šjƒ… …Šjƒ…pt‰…ƒrBuwB…Š‡3wBgdugp†v…Š™„¡‡y†¡G…Šnv†©ˆkt‰…p† Y
jƒ‚gs™f³kŽ‘t‰… ‡Œ—fu…EˆYgsjƒ‚‰‡…‘rŠ™fx
rŠ‚‰u¶gv†pgppkwBgdˆktvuwBe~†vg x€³k…Šqp†mwBgƒu†t‰…D‡{—vu… tdxziƒ…Šypc”¦‹T…Šn ϕ∗([α ∗ ¯β]) p [p∗(ψ)]
§¶…Bugp†p…ƒ™„¡TsVˆY‚‰uy%t‰…Šqsy ψ
wBgƒu γ ∗ ε
ˆkt„…v† Y
§6¡Tg rƒ™fxzrŠ‚‰uG†pg~x€³k…Šqv†¥t„…‘‡{—vu…mtfxziƒ…ƒy§6—fgpidgd—fŽ
γ ∗ ε(1) = y = ε(1)
U”didx€rƒ‚tˆY³kŽdjdg~bsc{w
c| sV™fgt„… ε
‚‰‡†pgduTgv†p—Š™„¡‡yVˆ‡gt‰…Šq ϕ ◦ ¯β
jd‚hgd™d³YŽat‰… γ(1)
wBgƒuYtfxziƒ…ƒyt„… y
c ¡
‰Vtfˆ‡uTt‰… ¯ε
¡Gg¥‚‰‡†pgƒu‡gp†p—ƒ™„¡TyVˆYgat‰…Šq ϕ ◦ β
wBgƒuk¡Ggx€³k‚‰uTgd™d³YŽat‰… y
wŠgƒu‡tdxziƒ…Šytt„…¤ˆYgdjd‚„‡… γ(1)
c}h—f™f—Šjd…Šy δ
‚‰‡†vgƒuƒepisiƒgjƒ‡{gtfxzt‰…Bu{ggp†v—Š™„¡‡yVˆ‡gkcR$rŠ—%jd…p†vgd—vuwB—ptfgpt‰ggv†p—Š™„¡‡yVˆ‡gsy}—f™f—ƒjdy†† §
x€rƒ‚zt„gduk—stfu δ = ¯ε
wBgdu‡es™fg γ(1) = δ(1)
c♦
«†…R`† ¨ (`w¡ fw 12 gz”}• p : (Y, y) → (X, x)
‡‰„‘’ˆ“•”–…—‡™˜Ž„•“ ‚ …—‡ê8€f…gihkwx”–ex” ‚98

i yX€ σ ‚Bƒ€B„†…z‡– ‚ …|gz”xeprxl8€xl•“fy”–…™r ‚ vy‰gzh””Bl x
‡‰„†… ˜σ ‚Bƒ€B„†…ihŽrxlf€n„ p …s‡™˜„†€fe q 5f•%gzh
”Bl–’ σ
r ‚ „qt P˜ ”Bl y
w”–ex” ‚ ”Bl ˜σ mt ‚ …G” m ilfo ”Bl gihˆr ‚Bƒl y
„†€ ‡‰„†…0r3e8€xl8€ „†€
[σ] ∈ p∗(π1(Y, y))‚
ii yX€ σ, σ
w pVq e‘rxl8…‰gi”Bl8€ X
r ‚ „q tG˜ ”Bl”gzh™r ‚‘ƒl x
‡‰„k…‰” m ilfo”Bl x
‡k„k… ˜σ, ˜σ
l8…
„†€xl q 5v%g ‚ …|oX”Bl–’koXg ‚ pVq eBr‰l‘’koXgz”‘l8€ Y
r ‚ „qt P˜ ”Bl y
”xe–” ‚98
”n„ ˜σ, ˜σ mtAl–’†€ ”Bl
ƒp …|lj” m ilfop„•€‡‰„k…zr3e8€xl8€j„†€ [σ ∗ ¯σ] ∈ p∗(π1(Y, y))‚
(xw ( £¢
G-Coverings, Deck Transformations
§ Ù ¨ …RA (`wˆfw¥¤3…„ ‡‰y65
‚ p : Y → X
‡‰„‘‹’ˆ“†”‘…s‡™˜ „•“ ‚ …—‡ê8€f…gihkw‘’ˆ“fy qt ‚ …elBr‡y p „
Aut(Y/X)
h l“•l ƒ„ l8€xlBr‡y
‚ ”n„k…0l‘r‰y p„ A
‚Bƒ€n„k…GlBr‡y p „†r ‚ “q y “ h ”xh‹€ g ©†€@5
‚ gih g’€
€B„ q ”8˜™g ‚ •0€ ‚C
”¸•0€ Deck transformations
w‰˜ lBr‡y p „X”¸•0€ Covering transformations8
Aut(Y/X) = {ϕ : Y → Y : ϕ ‚Bƒ€B„†…vlBrxl8…|lBrxl qŠ …gr3e8oe‡‰„†… p ◦ ϕ = p} .

€ · ¢¡¤£¦¥¨§¦¥©¢¥¡ ¥©¢ ¡ ¥© !¥! ¡#£¦ Galois
u¦…sisiƒ…Š‡‡ˆYgdjdgp†vtfuwB…B‡BwBgsidqprƒtvuwB…Š‡k³BsV™f…ŠuTrŠ™f…ƒwŠnsrdt„…ƒqp†igprŠ—tdgp†%—f™fesˆ‡gjƒu{gdy…ƒjƒes—fgdy
G
ˆY‚Xxž†pgv†a³BsV™f… Y
jd‚ X
†vga—fgpids††p‚‰ukt„…v†%³ksV™d…¤t€y††‚tf™f…s³Yu{s††£tdgdy—d™fedˆYgsyhgdqptfŽsysc
§ ` ¨ …RA (`w (¡ 3w£¢ q y‰gzh r`…„koelBr‡y p „ko (G, ·)
g ‚pm €B„•€At0v q l Y
Ar„•“e „q …|gz” ‚ q y Cj‚Bƒ€B„†…
r ƒ„„•“ ‚ …s‡ê8€8…|gzh G × Y → Y : (g, y) → g · y
wf“l–’X…s‡‰„•€xl•“l8…‚‘ƒ ”n„„k‡êVil–’5•„ 8
i g · (h · y) = (g · h) · y, ∀ g, h ∈ G
‡‰„†… y ∈ Y,
ii idG · y = y, ∀ y ∈ Y,
iii ˆ „“ ‚ …—‡ê8€f…|gzh y → g(y) = g · y ‚Bƒ€n„k… m €n„‰opl‘rxl8…|lBrxl qŠ …|gr‡efod”Bl–’ Y, ∀ g ∈ G.
¡
‰Vtfˆ‡u6g G
…Š™v‡{f‚‰u jŠ‡{g¤…Šjƒes—fg¤gsrƒ—~…Šjd…Bu…Šjd…Š™f‹Guˆ‡jd…Šndyit‰…Šq¥³BsV™f…Šq Y
ci›‚nd…mˆYgsjƒ‚‰‡{g
y, y ∈ Y
iƒx€jd‚}—stfukgv†pŽpwB…Šqv†ˆktfgv†£‡{—vu{ghtf™f…s³Yu{e£gv†iqprŠes™f³k‚‰u g ∈ G : g(y) = y .
ehgp¡‡sVyg G
‚‰‡†vgƒuT…ƒjƒes—fg¨§Ygsqstd— ‚‰‡†pgduTjŠ‡{gwŠidedˆYguˆY…Š—fqv†pgsjŠ‡{gsysc@$†ˆ‡qsjs”ƒ…siƒ‡ˆky X = Y/G
†pg‚‰‡†vgƒuTt‰…~ˆYnv†p…sid… t€y††td™f…s³‡us†† §T—fgpiƒgs—fŽat‰…mˆYnp†p…pid…¤t€y††wŠidesˆ‡‚zy††¥uˆ‡…Š—dqp†pgsjŠ‡{gsy§Ytf—ptf‚jdrƒ…Š™ds †pg~…Š™f‡ˆYy tfgv† ‹TqsˆTuwBŽ¤rŠ™f…s”Š…piƒŽ p : Y → X
rƒ…Šq~gprŠ‚‰uwB…v†s‡{f‚‰u wŠep¡T‚‚ˆkt‰…Bu³k‚‰‡…t‰…Šq Y
ˆktfgv†£wŠidedˆYg¨§Š—fgsidgs—fŽ‚tdgp†£tf™d…d³Yu{e‚rƒ…Šq%t‰…arŠ‚€™vu{x€³k‚„uc¶h˜³BsV™f…Šy X
‚‰‡†vgƒuY‚‰‹‡…Š—vu{g
ˆ‡jdxž†p…Šy£jƒ‚htfgv†t‰…ƒrŠ…pid…sp‡‡{g%rŠgpiƒ‡{wŠ… §T—dgsidgd—dŽ¥wŠep¡T‚ U ⊂ X,
‚‰‡†pgdu6gv†p…BuwŠtd— t‰…Šq X
gv†f†t‰… p−1
(U)
‚‰‡†pgƒuGgv†p…BuwŠtd—¤t‰…Šq Y
cXh ˆYqdjp”Š…piŠuˆ‡jd—Šytfgsyi‹TqsˆTuwBŽsyhgdqptfŽsyrŠ™d…d”Š…pidŽdyjd‚t‰…¥pY™fesjƒjdg p
—f‚ž†%‚„‡†pgduYtdqd³kgd‡…ŠypcRehept‰y¬gprŠ—wBeprŠ…Bu{‚€ytrƒ™f…¥¤prŠ…d¡Tx€ˆ‡‚‰uy¦pTu{g%t‰…v†a³BsV™f… Y
wBgduYtfgv†‚—f™fesˆ‡g‚tfgsy…Šjƒes—fgsy G
§Yg p
‚‰‡†pgdu‡jŠ‡{g%wŠgsidqsrdtvuwBŽigprŠ‚‰uwB—p†puˆ‡gkc
§ ` ¨ …RA (xw (P(`w£¦
ƒ„”l‘r‰y p„ G
pVq „ evenly§
gz”Blf€At%v q l Y
w‰„†€X‡‰y5
‚ gihˆr ‚BƒlX”‘l‘’ Y
mt ‚ …`r`…„‘’ ‚ …”Blf€8…{y V
” m ”Bl8…{„ v%gz” ‚ g · V
‡‰„†… h · V ‚Bƒ€n„k… “†m €B„ ’‹…{„6‡‰y5
‚ p …{„ Š lq ‚ ”‘…s‡‰y
g, h ∈ G‚
$ ¨'¨ (`w (6‰w”y€ r ƒ„‘lBr‡y p„ pVq „ evenly
gz”Blf€dt%v q l Y
wP”xe–” ‚ h Š ’kg…—‡™˜ “q l xlsv˜
p : Y → Y/G ‚‘ƒ€n„k…zr ƒ„Ž‡‰„‘’ˆ“•”–…—‡™˜Ž„•“ ‚ …—‡ê8€f…gih ‚
y“•e p ‚ …“ h 8
• p
‚‰‡†vgƒuPjŠ‡{gmˆYqp†p‚€³kŽsy%wBgduPgv†p…BuwŠtdŽ~gprŠ‚‰uwB—p†puˆ‡g¤wBgv¡TsVy%p‡u{g~wBev¡T‚ V
gp†v…BuwŠtf—©t‰…Šq Y
x€³k…Šqsjƒ‚%—ptvuPt„… p−1
(p(V )) = ∪g∈Gg · V
‚„‡†pgdugv†p…BuwŠtd—‘t‰…Šq Y
ˆ‡gv†xž†vyVˆ‡gigv†p…BuwŠt„s††XwBgduƒgprŠ—it‰…p†…Š™vuˆYjƒ—itfgsy}gsrƒ‚‰u{wŠ—p†suˆ‡gsy'rŠgpiƒ‡{wŠ…it‰… p(V )
‚„‡†pgduƒgv†p…BuwŠtd—t‰…Šq Y/G
ct| sV™fg gv†rƒed™f…ƒqdjd‚t„… V
§G—ƒrƒyVy£ˆYt‰…v†¥…Š™vuˆ‡jd— tfgsy even
—f™fesˆ‡gsy§Tt„g g · V
¡Ggi‚‰‡†pgduk“dxz†vg£jd‚zt„gs“fnht‰…Šqsysc3t™„wB‚‰‡Š†vg£—d‚„‡{“f…ƒqdjd‚}—stfuBp‡u{ghwBev¡G‚ tdx€t‰…Bu… V
t‰… p(V )
‚„‡†pgdu…ŠjƒgpidewBgsidqsjƒjƒxž†p…igprŠ—itfgv† p
§Š—fgpidgd—fŽX…‚rŠ‚€™vu…Š™vuˆYjƒ—Šy tdgdy p
ˆY‚ wBev¡G‚}xž†pghgprŠ—it‰g g ·V
ˆYt‰… p(V )
‚‰‡†pgƒuYxž†pgdyX…Šjƒ…Bu…Šjƒ…Š™d‹GuˆYjƒ—Šypcut™despYjƒgptvu}p‡u{g y ∈ V
§—dgsidgd—dŽ p(y) ∈ p(V )
qprŠes™f³k‚‰u g · y ∈ g · V
xztfˆ‡u}sVˆYtd‚
p(g · y) = p(y)
wBgduBed™dg%‚‰‡†pgƒuk‚zrB‡c†‰VrŠ‡ˆ‡gsyXgv† p(g · y1) = p(g · y2)
qsrƒed™f³k‚‰u h ∈ G
jƒ‚
h · g · y1 = g · y2
cR¦‹T…ƒn%g£—f™fesˆ‡g£‚‰‡†vgƒu even
§ƒwBgduBˆ‡qv†p‚zrƒsVy¦‚ziƒ‚€nv¡G‚€™fghgprŠ—‚t„…v†i…Š™fuˆ‡jd—jƒgsy§Š¡TgarŠ™dx€rƒ‚‰u h = idG
wŠgƒuYed™fg‚gfgpt„…ƒndjd‚z†vg‚gprŠ‚‰uwB—v†suˆYg%‚‰‡†pgƒu6b bdc♦
§ ` ¨ …RA (`w (G¦3w 3
y65
‚ ‡k„–‹’ˆ“†”‘…s‡™˜Ž„•“ ‚ …s‡ê8€8…|gzh p : Y → X
w‡“•l–’ “q l m qt ‚ ”n„k…„•“edr ƒ„
even
pVq ykgih6r ƒ„kodlBr‡y p„ ko G
g ‚ m €n„†€Gt0v q l Y
l8€xlBr‡y
‚ ”n„k… G
€s‡‰„‘’ˆ“•”–…—‡™˜ „•“ ‚ …s‡ê8€8…|gzh
‡‰„†…vl%t0v q lfo Y G
€s‡‰„‘’ˆ“•”–…—‡êfo3t%v q lfo ‚
¨ µ–Ÿsń˜®8x i¬¢ T@T ´}™•ŸV ›d™}¤B™•§s´}}¤}šV™‘˜`¬ŸP™B¢nšXIn¤ šWVn¤jž}´XaB¬@Vµ–ŸAšV}¤X¶–´} properly discontinuous© I T ¤UI‘¬•˜™Ž™B¬¢}¤nŸs°f£›¨± ¦‘§s¡WV
discontinuous
¬@V‘µx™‘ ¤BŸV˜‰¶nš}˜xx˜ š¸´}B°f˜§s›jŸV ¤B™‘˜`žn˜™n®xń˜šXa © µ–ŸešWVn¤šV–œ–n¦‘ T ˜®¤IP§ ¤¸¤Bx˜™dQ•žWVB¦‘™BžXIPž}Ÿ ¤0§s°f–¢}¤G¬@V‘µxŸV ™T¬¢B¬¬£´}Ÿs¢B¬@V‘›u¬fšV}¤ Y
¢nœ––¬ £}¤Bn¦‘™GšXV‘› Y
± ¦‘§s¡WV
properly
¬@V‘µ–™‘ ¤BŸV˜f¶nš}˜x®¤a Y†Ÿx¬¢Bµ‘œ–™ T I%¬ £}¤Bn¦‘0š¸§sµ‘¤nŸV˜8µ–¶}¤B0œ–Ÿ¨œ–Ÿs´}™B¬µ–§ ¤BG™Bń˜Y†µ–¶0™nœ–¶%š}˜›xµ–Ÿ¨šV™•x´¸§s›šVx¢8± ¶–´}–›v¶–µ› properly discontinuous
¬¢B°–¤Paxœ–™B´¸™†´XaPc}Ÿ¨šV™‘˜B®8™‘ñ¶nšV™}¤x°f´WV‘¬•˜µ––œ–x˜ŸV šV™‘˜B¬@V‘µx™‘ ¤BŸV˜¶nšn˜†®da Y†Ÿu¬@V‘µxŸV R§s°fŸV˜ T ŸV˜šV}¤‘ã
V
œ–x¢T¤n™Gš¸§sµB¤BŸV˜ˆµ–¶}¤BRœ–Ÿ¨œ–Ÿs´}™B¬µ–§ ¤BŸs›TšVPœ‘¦PIUY†x›Tµ–Ÿ¨šV™•x´¸§s› g · V
Q
šVx¢ V
±`¤u¬Ÿx™B¢nšWIn¤ušXVn¤%Ÿs´¸µ9Vn¤nŸV ™%œ–´¸x¬dY†§s¬–¢BµxŸ‡šWVn¤f¦‘§s¡XV! Ÿ¨¦‘Ÿ £ Y†Ÿs´}™ Qš¸¶nš}Ÿ!V
even
ž¸´ aB¬@V0µx™B›UQ9Y†™¬@V‘µx™‘ ¤BŸV˜ freely and properly discontinuously
±

¡ ‘ !¡9£¦¥¨§¦¥©!#¥ ¡¦¥©¢# ¡ ¥ ·
¸
«”†`335Y†ù†uu (`w ( ¯ wGyX€Plxt0v q lfo Y ‚‘ƒ€n„k… Hausdorff
‡‰„†…k„•€RhX“ ‚ “ ‚ q „kg‹r m €•h™o%”–y “ h™o
l‘r‰y p„ G
pVq „ ‚ ‚ ©5
‚ q „ g ‚ „‰’‰”xe8€–w p h8k„ p ˜ g ∈ G {idG} : g(y) = y, ∀ y ∈ Y
w ”xe–” ‚h G
pVq „ evenly
gz”‘l8€ Y ‚
q y8’rv„”‘…8
²u{gpB†vyVˆktf— y ∈ Y
§ ‚zrBuiƒxzpky “dxz†v‚‰yipY‚‰ut‰…p†su{x€y Ug
t‰…Šq g · y ∈ Y
§Gjƒ‡ŒgpTu{gXwŠep¡T‚@xž†pg g ∈ G
8gtnsrƒgd™d“fgXgsqst„s††¦‚€“fgdˆY‹Tgpiƒ‡Œd‚€t‰gdufgsrƒ—tfgv† Hausdorff
ˆYqv†f¡GŽpwBgkc
cxXx€t€y
V =
g∈G
g−1
· Ug
jƒ‚ V
gv†p…BuwŠtf—‚t„…ƒq Y
rƒ‚€rƒ‚€™fgdˆYjƒxž†pgit‰…ŠjƒŽ£gv†p…BuwŠtds††£‚‰‡†pgduBgp†p…Šu{wƒtf—aˆYnp†v…sid…
§ƒ†pg£‚„‡†pgdup‡‚‰ut‰…v†su{eXt‰…Šq y
c|}—std‚'p‡u{gXwBev¡G‚@—vu{gd‹‡…Š™f‚ztvuwBe g1, g2 ∈ G
§p¡Gghx€³By —ptvust„… g1 ·V ∩g2 ·V
uˆ‡…ƒnst‰gduYjd‚
g∈G{g1}
g1 ·g−1
·Ug ∩(g1 ·g−1
1 ·Ug1 )∩(g2 ·g−1
2 ·Ug2 )∩
g∈G{g2}
g2 ·g−1
·Ug = ∅.
²u{gptv‡ Ug1 ∩ Ug2 = ∅.♦
«”`x…P5k†uu (`w ( ° w 12 gi”¸• p : (Y, y) → (X, x) G
€s‡‰„–‹’ˆ“•”–…s‡ˆ˜ „•“ ‚ …—‡ê8€f…gih ‡‰„†… Y
€n„
‚‘ƒ€n„k…ig’†€ ‚ ‡™”‘…s‡êfo3t0v q lfo ‚¢¡ ex” ‚ h G ∼= Aut(Y/X)‚
yX“e p ‚ …“ h 8
l@t‰gp¡T‚€™f…ƒrƒ…Bus xž†pg y ∈ Y
c@²u{gapB†vyVˆktf— ϕ ∈ Aut(Y/X)
§Yt‰… y
wBgdu‡t‰…
ϕ(y)
¡Gg‚rŠgpp‡gd‡†p…Šqv†ijƒx€ˆky tdgdyt‹TqsˆTuwBŽsytrƒ™f…d”ƒ…sidŽdy$ˆktfgv†%‡Œ—fuŒg£tf™f…s³Yu{e‚rŠ…ƒq‚t‰g‚rƒ‚‰™fuŒx€³k‚‰ucl'qp†p‚zrdsVy£¡Tgmqsrƒed™d³Y‚‰u xž†pg g ∈ G : g · y = g(y) = ϕ(y)
c‚| sV™fg©uˆ‡³kns‚„u p ◦ ϕ(y) =
p ◦ g(y) = p(y)
cV$rŠ—¥iƒŽsjƒjdg bsc´¨§k…Bu ϕ
wBgdu‡g g
¡TgaˆYqdjsrB‡rƒt‰…Šqv†Bc♦
§ Ù ¨ …RA (`w ( ¥fw 12 gz”¸• p1 : (Y1, y1) → (X, x)
‡‰„†… p2 : (Y2, y2) → (X, x)
‡‰„–‹’€
“†”‘…s‡ m oj„•“ ‚ …s‡ˆl8€ ƒg ‚ …|oX”Bl–’jt0v q l–’ X ‚ 12 €B„koŽlBr‰lBrxl qŠ …|gr‡efoŽ”Bl–’ t0v q l–’ (Y1, y1)
gz”‘l8€
t0v q l (Y2, y2) ‚Bƒ€B„†…ˆr ƒ„ gv’k€ ‚tG˜™oA„•“ ‚ …—‡ê8€f…gih ϕ : Y1 → Y2
” m ”Bl8…{„pv%gz” ‚ ”Ble“„ q „k‡‰y”¸•
p …{y8’ q „–r‡rv„ €B„ ‚Bƒ€n„k…r ‚ ”n„65
‚ ”–…s‡‰e 8
(Y1, y1)
ϕ
−→ (Y2, y2)
p1 p2
(X, x)
y€ph ϕ ‚Bƒ€B„†… m €n„‰o0l‘rxl8…|lBrxl qŠ …|gr‡efoT”¸•0€p”Bl•“lsvlf’‹…s‡•v0€xt%v q •%€ Y1
‡‰„†… Y2
w”–e–” ‚ “q lf‡ ©€
“†” ‚ … m €B„ko3…|g™lBr‰l q‹Š …|g‹r3efo3‡‰„‘‹’ˆ“†”‘…s‡†v%€it0v q •0€ ‚¤£R‚ „‰’‰”8˜™€0”8h‹€A“ ‚ q ƒ“•”¸•%gihRl8…•‡‰„‘’ˆ“•”–…—‡ˆl ƒt0v q l8… l8€xlBr‡y¡}l8€”B„†…i…gzeBrxl qŠ lx… A¨˜ …|g™l p ©†€n„xrxl8…Cf‚
l'gp†‚ˆYqp†pxzrƒ‚„u{g£t„…ƒq%iƒŽsjƒjdgst‰…Šy£bdc´#§Š¡Tgax€³By
«†…R`† ¨ (`w ( » w 12 gz”¸• ϕ1, ϕ0
lBrxl‘rxl qŠ …|g‹r‰l ƒ ”‘l‘’xt%v q l‘’ (Y1, y1)
gz”‘l8€t%v q l (Y2, y2)‚y€”’ˆ“fy qt ‚ … y ∈ Y1 : ϕ1(y) = ϕ0(y)
w‡”xe–” ‚ ϕ1 = ϕ0 ‚
ut™feppYjƒgptvuTgv† ϕ1(y) = ϕ0(y)
§‡¡Tg¥‚„‡³Yg%—stfu p2 ◦ ϕ1(y) = p2 ◦ ϕ0(y) = p1(y)
wBgdut„…¤gprŠ…ptfxziƒ‚€ˆYjƒg£xzrŠ‚zt‰gƒukedjd‚€ˆ‡gagprŠ—t‰…iƒŽsjƒjdg bscÝc
«†…R`† ¨ (`w ( 3w ˆ l‘r‰y p„ ””¸•0€ Aut(Y/X)
psq „ ‚ ‚ ©#5
‚ q „Xgi”Bl8€Tt%v q l Y ‚
¡
‰¶ˆYt€y p : (Y, y) → (X, x)
wBgpidqsrdtvuwBŽmgprŠ‚‰uwB—v†suˆYg ¡
‰¶ˆYt€y ϕ ∈ Aut(Y/X)
wBgdu
y ∈ Y
jƒ‚ ϕ(y) = y
c†|}—std‚ p2 ◦ ϕ(y) = p2(y)
wŠgƒuYed™fg ϕ = id
c
$ ¨'¨ (xw (Tˆfw 12 gz”¸• (Y1, p1), (Y2, p2)
‡‰„‘’ˆ“•”–…—‡ˆl ƒ t%v q l8…%”Bl–’ X
‡‰„†… yi ∈ Yi
wAr ‚
{i = 1, 2}
wT€n„ ‚Bƒ€B„†…‰gzhˆr ‚Bƒ„ m ”xg‹… v%gz” ‚¤8 p1(y1) = p2(y2)‚†Œ “fy qt ‚ …`l‘rxlBr‰l qŠ …|g‹r3efo
ϕ : (Y1, y1) → (Y2, y2)
„†€€ p1∗ (π1(Y1, y1)) ⊂ p2∗ (π1(Y2, y2))‚

š · ¢¡¤£¦¥¨§¦¥©¢¥¡ ¥©¢ ¡ ¥© !¥! ¡#£¦ Galois
}†aqprŠes™f³k‚‰uBtdxzt„…Šu… ϕ
§B¡Gg%x‰³By¬—stfu p2 ◦ ϕ = p1
§k—dgsidgd—dŽ%g ϕ
‚‰‡†pgƒukgv†p—Š™„¡‡yVˆ‡g‚tfgsy
p1
cV$rŠ—t‰…¡T‚zsV™fgsjƒgmbsc¡ £xzrƒ‚€t‰gdukt‰…¥fgpt‰…Šndjd‚ž†p…¥gsrƒ…stdx€id‚€ˆ‡jdgYc
«†…R`† ¨ (`w„ fw ¦
‚ ”–…|o6“ q l£¢‰“l@5
m g ‚ …o ”Bl–’ “q lfh‰’‹l©xr ‚ €xl–’ ˜ˆr‡ri„”Blfonwj’ˆ“fy q t ‚ …ul–€
r‰lx…|lBr‰l q‹Š …|g‹r3efo ϕ : (Y1, y1) → (Y2, y2)
wx„†€€ p1∗ (π1(Y1, y1)) = p2∗ (π1(Y2, y2))‚
xXgax€³k…Šqsjƒ‚X—ptvu
p2∗ (π1(Y2, y2)) ∼= π1(Y2, y2) ∼= ¤ π1(Y1, y1) ∼= p1∗ (π1(Y1, y1)).
l'gp†‚jƒ‡Œg%‚„u{—vuwBŽ‚rŠ‚€™v‡rƒt€yVˆYg%gsqst‰…Šn©§ŠrŠgd‡{™„†p…Šqsjƒ‚$t‰…rŠgs™fgswŠest€y gsrƒ…stdxziƒ‚€ˆYjƒg
«†…R`† ¨ (`w„†(`wjy€ (Y, p)
‡k„–‹’ˆ“†”‘…s‡ê8o3t%v q l8oX”Bl–’ X
wxr ‚ y1, y2 ∈ p−1
(x)
‡‰„†… x ∈
X
w”xe–” ‚ ’ˆ“y qt ‚ … ϕ ∈ Aut(Y/X)
r ‚ ϕ(y1) = y2
w‹„•€€ p∗(π1(Y, y1)) = p∗(π1(Y, y2))‚WEF‡%ù ¨ (`w„3‰w ¢ ©klj‡‰„‘’ˆ“•”–…—‡ˆl ƒ t%v q lx… (Y1, p1)
‡‰„†… (Y2, p2)
”‘l‘’Pt%v q l–’ X
w ‚Bƒ€B„†…
…|ge‘rxl qŠ l8… „†€€ ’‹…{„ ‡‰y65
‚ p ©†l y1 ∈ Y1
‡‰„†… y2 ∈ Y2
r ‚ p1(y1) = p2(y2) = x ∈ X
lx…
’ˆ“•l8l‘r‰y p ‚ o p1∗ (π1(Y1, y1))
‡k„k… p2∗ (π1(Y2, y2))
„•€•˜™‡ˆl‘’k€dgi”xh‹€ ƒp …{„X‡–iykgihŽgv’¡‘’’ ƒ„ko
gi”Bl8€ π1(X, x)‚
ehgv¡TsVy}…Bud—fnd…iwBgpiƒqprƒtfuwB…B‡v³BsV™f…BuŠ‚‰‡†pgƒuƒuˆ‡—ƒjƒ…Š™f‹‡…Bu§ƒgprŠ—hrƒ—Š™vuˆYjƒg%bsc{w ©§p¡Tghx€³By”—stfu
p1∗ (π1(Y1, y1)) = p2∗ (π1(Y2, y2))
c©| sV™fg y1, y2 ∈ p−1
(x)
wBgƒuPgsrƒ—‘¡T‚zsV™fgsjƒg”bdc·¨§
xzrŠ‚zt‰gƒuBt‰…¤dgst‰…Šnsjƒ‚ž†p…Tc|}…¤¡T‚zsV™fgdjdg jdgdyhidx€‚„u‡—stfu6gwŠidesˆ‡gˆYqddqsp‡‡{gdyht€y††qprŠ…ƒ…Šjƒes—dy††arƒ…Šq¥gp†vgd‹‡x‰™d‚€t‰gduTˆkt„…¡G‚zsV™fgsjƒg bdc·¨§BwBgv¡T…Š™v‡{f‚‰u‡gprŠ—piƒqpt‰ght„…v†‚wBgsidqprƒtvuwB—%³ksV™d… up to isomorphism.
$ ¨'¨ (xw„ ¦3w 12 gi”¸• ‡‰„‘’ˆ“•”–…—‡ˆl ƒ t%v q l8… (Y1, p1)
‡‰„†… (Y2, p2)
”Bl–’et%v q l–’ X
‡‰„†… ϕ
m €B„koplBr‰lBrxl qŠ …|gr‡efop„†€8y–r ‚ g‹y”Bl–’ko ‚ ¡ e–” ‚ l (Y1, ϕ) ‚Bƒ€B„†…‡‰„‘’ˆ“•”–…—‡êfo‰t0v q lfo ”Bl–’ Y2 ‚
y“•e p ‚ …“ h 8
ehep¡T‚©ˆ‡gsjƒ‚‰‡… x ∈ X
x€³Y‚‰u jƒ‡ŒgD—f™d…Šjƒ…ŠˆYqv†p‚zwŠtvuwBŽ”pY‚‰ut‰…p†su{e U
§$xztfˆ‡usVˆYtd‚¥†vg ‚‰‡†pgƒuR…Šjdgside©wBgpiƒqsjƒjdxz†vg‘wŠgƒuRgprŠ— tfuy~—fns… wBgpidqsrdtvuwBx€ygsrƒ‚‰u{wŠ…p†s‡ˆ‡‚‰uy p1, p2
t‰gdqptf—s³k™f…p†vg'$† U1
‚„‡†pgduG…ŠjdgsideawBgpiƒqsjƒjdxz†vggsrƒ— tfgv† p1
wŠgƒu U2
…ŠjƒgpiƒewBgpidqdjdjƒxž†pggsrƒ—tfgv† p2
tf—ptf‚$¡Gxzt‰…v†vt‰gdy U = U1 ∩ U2
§B¡Tg%rŠ™f…dwBn%‡‚‰u‡gafgpt‰…Šnsjƒ‚ž†pg‚pY‚‰ut‰…v†su{eYc „ p ‚Bƒ|“ • ex”‘…‡h ϕ ‚Bƒ€B„†… ‚ “ ƒ8
}† y ∈ Y2
¡Gg¤—f‚‰‡{“dy —ptvu qsrƒed™d³Y‚‰u x ∈ Y1 : ϕ(x) = y.
‰VrBuiƒxzpky©”ŠesˆYg y1 ∈ Y1
wBgdu y2 = ϕ(y1)
jd‚ p1(y1) = p2(y2) = x
c†utgd‡{™„†vy f : I → Y2
jƒ‚¥gs™f³kŽ‘t‰… y2
wBgdutfxzid…Šyt„… ˆ‡gsjƒ‚‰‡… y
c ¡
‰¶ˆYt€y g = p2 ◦ f
†vgE‚‰‡†pgdu†gE‚‰uwB—v†pg©t„…ƒqrŠgs™fgprŠev†vy —d™f—Šjƒ…ƒqhjƒx€ˆYy”tdgdy p2
ˆYt‰…v†³BsV™f… X
§djƒ‚'gd™f³kŽt‰…£ˆ‡gsjƒ‚‰‡… x
c†¦rƒ—‚t‰…£idŽsjƒjƒggp†v—Š™„¡‡yVˆ‡gsy—f™f—Šjsy†† §RqprŠes™f³k‚‰u3jƒ…v†pgs—vuwBŽmgv†p—Š™„¡‡yVˆYg’ˆY‚a—f™d—Šjƒ… h
ˆkt‰…p†m³BsV™f… Y1
§jƒ‚gd™d³YŽ£t‰…¤ˆYgsjƒ‚‰‡… y1
§Btfxzt‰…Bu{g‚sVˆktf‚ p1 ◦ h = g
c ¡
‰¶ˆYt€y¬†vg%x€³k‚„uktfxzid…Šytt‰…ˆYgdjd‚„‡… x
c@xXg—f‚‰‡{“dy —stfu ϕ(x) = y.
huG—f™f—ƒjƒ…Bu ϕ ◦ h
wŠgƒu f
x‰³k…Šqv†atdgp†¥‡Œ—fuŒg¥gs™f³kŽ¨§Yt‰… y2
wBgdu uˆY³Yns‚‰u
p2 ◦ ϕ ◦ h = p1 ◦ h = g = p2 ◦ f =⇒ ϕ ◦ h = f
§Pgsrƒ—©jƒ…v†pgs—vuwB—stdgst‰g¤gp†v—Š™„¡‡yVˆ‡gsy—f™f—Šjsy††Bc ¡
‰Vtfˆ‡u ϕ(x) = y
c¡
‰Vtfˆ‡uf—vu{gsidxzpY…Šqdjd‚…Šjdgside'wBgpidqdjdjƒxž†pg@rƒ‚€™vu…s³YŽ@t„…ƒq tfqs³Ygd‡…Šq z ∈ Y2
yVyV‚€“fŽsy rƒgƒ‡{™„†p…ƒqdjd‚
U
gp†p…Šu{wƒtfŽirŠ‚€™vu…s³kŽit‰…Šq x = p2(z)
§Šg%…ƒrƒ…B‡{g£‚‰‡†pgduB…ŠjƒgpiƒehwBgpidqdjdjƒxž†pghwBgdukgprŠ—‚tvuy¦—fns…wBgpiƒqprƒtfu{wŠx‰ytgsrƒ‚‰u{wŠ…p†s‡ˆ‡‚‰uyjd‚ht„…v†atd™f—ƒrƒ… rŠ…ƒqarŠ‚€™vupY™fe%Tgsjƒ‚ˆktfgv†¥gs™f³kŽYc$xXx€t€y W
§Y†pg‚‰‡†vgƒu t‰…©‹Tnpidid…~tfgsy p−1
2 (U)
rŠ…ƒq¤rƒ‚‰™fuŒx€³k‚‰uGt„… z
c£• W
‚„‡†pgdu6…Šjƒgpiƒe¥wBgpiƒqsjƒjdxž†pg¥gsrƒ—tfgv† ϕ
c ♦¡
‰¶ˆkt‰y (Y, p)
wBgpiƒqprƒtfuwB—ŠyR³BsV™f…Šy¶t„…ƒq X
§vjd‚ Y
gprƒide}ˆ‡qv†p‚zwŠtvuwB—X³ksV™d…Tc†}† (Y , p )
‚‰‡†vgƒu6xž†pg tdqd³kgd‡… § —vu{gs‹T…Š™d‚€tfuwB—~wBepiƒqsjƒjdg¤t‰…Šq X
§ tf—ptf‚£gsrƒ—mt‰…miƒŽsjƒjdg’bsc‘b¦¥¨§Gqsrƒed™f³k‚‰u…Šjƒ…ƒjƒ…Š™f‹TuˆYjƒ—Šy ϕ
t‰…Šq (Y, p)
ˆkt‰…p† (Y , p )
wBgduVgsrƒ— t„…”rƒgd™fgprŠev†vy˜idŽdjdjƒgE… (Y, ϕ)
‚‰‡†vgƒu xž†pgsy‚wBgpiƒqprƒtfuwB—Šyi³BsV™f…Šy‚t‰…Šq Y
c£tqptf—ƒy%‚‰‡†pgƒu …mid—spY…ŠyirŠ…ƒq~xž†pgsy‚gprƒide ˆYqv†p‚
wŠtvuwB—ƒyiwŠgsidqsrdtvuwB—ŠyX³ksV™d…Šy%…v†p…Šjdedf‚zt‰gdu‡wBgv¡G…piŠuwB—ƒywBgsidqprƒtvuwB—ƒyh³ksV™d…Šysc¦‰VrB‡ˆYgsy£gprŠ—t‰…¡T‚zsV™fgsjƒg~bsc{wsw‚wBev¡G‚t—fns…¥wŠgp¡T…siƒuwB…B‡ŠwBgpiƒqprƒtfuwB…B‡B³BsV™f…BuT‚‰‡†pgduTuˆY—Šjƒ…ƒ™f‹T…Buc
§ §¤ n™B›fxµ–8˜–µx–´ †˜¬µ–¶x›3Ÿ¨œ9a T ŸV˜k˜¬–µ–x´k˜¬µ–¶Aš ‹¤0Ÿsµ–œ‘¦‘Ÿ¨®x¶xµ–Ÿ ¤ ™¤ Y†Ÿsµ–Ÿ¨¦–˜ž}£™¤A–µ¤aBž‹¤x±

¡ ‘ !¡9£¦¥¨§¦¥©!#¥ ¡¦¥©¢# ¡ ¥ · o
(xw ( º ¢¡ qªT`3…†fu 5u†A §‘¨ …†ªG†A π1(X, x)
'5kH£¢¥¤PIHP13H p−1
(x)
¡
‰¶ˆYt€y p : (Y, y) → (X, x)
wBgsidqprƒtvuwBŽ'gsrƒ‚„uwB—v†suˆYg©§d…Š™f‡Œ„yE—d™fedˆYg}tfgsy@…ƒjƒes—fgdy π1(X, x)
ˆYt‰…mˆYnp†v…sid… p−1
(x)
pTu{g¥wBev¡G‚ x ∈ X
xztfˆ‡uTsVˆYtd‚‚g π1(X, x)
†pg¤—f™fg¤gsrƒ—~—f‚€“vu{e¥ˆkt„…ˆ‡nv†p…piƒ… p−1
(x)
c
§ Ù ¨ …RA (xw„ ¯ w 12 gz”}• (Y, p)
‡‰„–‹’ˆ“•”–…s‡‰efoet%v q lfo ”Bl–’ X
wpr ‚ x ∈ X ‚ ¤3…{„ ‡‰y5
‚
y ∈ p−1
(x)
‡‰„†…‰’‹…{„ ‡ky65
‚ [α] ∈ π1(X, x)
wul q ƒs• y · [α] ∈ p−1
(x)
•%o ‚n“ ˜‹o ‚ yX“eŽ”Bl
˜ˆr‡rv„ „†€fe q 5f•%gzh‹o pVq eBr•0€‡‰„†…`”‘l 5
‚ v q hˆrv„ ~ ‚§¦ w0’ˆ“fy qt ‚ …vrxl8€B„ p …s‡™˜6‡–iykgih psq eBr •%€
”‘l‘’ Y
w m gz”}• [˜α]
w‰” m ”‘lx…„ v%gi” ‚ p∗([˜α]) = [α]
r ‚ „qt P˜ ”Bljgzhˆr ‚Bƒl y‚ §
q ƒs• y · [α]
€n„
‚‘ƒ€n„k…v”BlX” m ilfod”xh™oe‡xzy‰gzh™o pVq eBr•0€ [˜α]
w p h8k„ p ˜ y · [α] := [˜α](1)‚
•–ˆ‡³kns…Šqp†£t‰ga‚€“fŽsy
(y · [α]) · [β] = y([α] · [β])
wBgdu y · [ex]
t©¨ = y.
l'qdjsrŠ‚€™fgd‡†v…Šqdjd‚¥—stfu¶g π1(X, x)
gsrƒ— —f‚€“vu{e ˆkt‰…”ˆ‡nv†p…piƒ… p−1
(x)
c xXg’—f‚‰‡{“f…Šqsjƒ‚¤—stfug’—d™fedˆYg©gsqstdŽm‚‰‡†vgƒujd‚zt„gv”ƒgstfu{wŠŽ¨§6—fgpidgd—fŽ pTu{gmwBev¡T‚ y1, y0 ∈ p−1
(x)
qsrƒed™d³Y‚‰u [α] ∈
π1(X, x) : y0 · [α] = y1.
u¦™fespYjdgstfu wŠgp¡‡sVy%x€³k…Šqsjƒ‚iqsrƒ…d¡Tx‰ˆY‚‰u —stfuP…©³BsV™f…Šy Y
‚‰‡†pgdu—f™f…Šjd…Šˆ‡qv†p‚zwŠtfu{wŠ—Šy§¦qprŠes™f³k‚‰u@wŠidedˆYgD—f™f—Šjsy†† [˜α]
gsrƒ—¬t‰… y0
ˆkt‰… y1
c xXxzt€y [α] =
p∗([˜α])
cV|}… [α]
‚‰‡†pgduYwŠidedˆYg‚wƒiƒ‚‰uˆktds††‚jƒ…v†p…ƒrƒgstfus††%wBgdu y0 · [α] = y1.
l@tfgv†aˆYqp†vx‰³k‚‰u{g%¡Ggrƒgd™f…ƒqdˆ‡u{edˆY…Šqsjƒ‚$wBesrƒ…Bu{g‚”ƒgdˆ‡uwBeagprŠ…ptf‚ziƒx€ˆYjƒgpt‰g‚wBgduT…Š™vuˆ‡jd…Šndygsrƒ—¥tdgp†i¡T‚zyV™v‡{g…Šjƒes—dy††‚rŠ…Šqi¡Ggjdgdyt”ƒ…Šgv¡GŽsˆY…Šqp†h†vg%rƒidgdˆ‡u{edˆY…Šqsjƒ‚$t‰…p†‚ˆkwB…ƒrŠ—¥jdgdypc¡
htt‰gp†¤jƒ‡Œg …Šjƒes—fg G
—f™dg~ˆY‚‚xž†pg ˆ‡nv†p…piƒ… E
gprŠ—©gs™vuˆktf‚€™fe©§Gtf—ptf‚£iƒx€jƒ‚i—ptvu6… E
‚‰‡†pgduxz†vgdy£gd™fuˆYtd‚€™f—Šy G
³BsV™f…Šypct$†¥g G
—f™dg jd‚€t‰gv”ŠgptvuwBe%ˆYt‰…v† E
§Gtf—ptf‚i… E
…v†p…Šjdedf‚zt‰gdu…Šjƒ…ppY‚z†vŽdy
homogeneous G
³ksV™d…Šysc¦rƒ—¦t‰…v†'td™f—ƒrƒ…trŠ…ƒq …Š™v‡ˆYgsjƒ‚tdgp†'—f™fesˆ‡g}…ƒjƒes—fgdy8ˆYqp†d¡GŽpwBg~±
rƒ™f…ƒwBnprƒtd‚‰u6—ptvu6g~gprŠ‚‰uwB—v†suˆYg E → E
rŠ…ƒq~gprŠ‚‰uwB…p†p‡Œd‚„u y → g · y
‚‰‡†pgduxz†vgdy…ƒjƒ…Bu…ƒjƒ…Š™f‹TuˆYjƒ—ŠyXt‰…Šq E
wBgduYˆYqp†v‚€rdsVyhjŠu{gajd‚€tdep¡T‚€ˆ‡g£t‰…Šq E
c ¡
‰Vtfˆ‡uTx€³k…Šqsjƒ‚¦
WEF‡a`vu ¨ (`w„ ° wyX€ E ‚Bƒ€B„†… m €n„‰oX„ q …|gz” ‚ q e8o G
€ƒt%v q lfoBwG”xe–” ‚ ’‹…{„ ‡‰y5
‚ g ∈ G
wAh
„•“ ‚ …—‡ê8€f…gih E → E
“l–’„“ ‚ …—‡ˆl8€ ƒ
‚ … y → g · y ‚Bƒ€B„†…zr`…{„r ‚ ”–y5
‚ gih ”Bl–’ E ‚
§ Ù ¨ …RA (xw„ ¥3w 12 gi”¸• E1, E2
wP€n„ ‚Bƒ€B„†…x„ q …|gz” ‚ q l ƒ G
€ƒt%v q lx…‚ ¦
ƒ„6„“ ‚ …—‡ê8€f…|gzh f :
E1 → E2
l8€xl‘r‰y
‚ ”B„†…„•“ ‚ …—‡ê8€f…gih”„ q …|gz” ‚ q v0€ G
€t0v q •%€ A¨˜ G-equivariantC
„•€
f(g · y) = g · (fy)
’‹…{„ ‡ky65
‚ g ∈ G
‡‰„†… y ∈ E ‚ ¦
ƒ„†„•“ ‚ …s‡ê8€8…|gih f
„q …gi” ‚ q v0€ G
€t0v q •%€†‡k„– ‚Bƒ”n„†…
…|g™lBr‰l q‹Š …|g‹r3efo„ q …|gi” ‚ q v%€ G
€ƒt%v q •0€–wA„•€ h f ‚Bƒ€B„†…3~8€~ ‡‰„k… ‚ “†…w%r ‚ ”8h™€”„†€” ƒgz” q l Š ˜
”8h™oBwf€n„ ‚‘ƒ€n„k…i‡‰„†…z„‰’‰”x˜ r ƒ„„“ ‚ …s‡‰ef€8…|gzh „q …|gi” ‚ q v%€ G
€ƒt%v q •0€ t}t ‚
¡
‰¶ˆYt€y E
§†xž†pgsyatfqs³kgƒ‡…Šya…Šjd…spY‚ž†pŽsyagd™fuˆYtd‚€™f—Šy G
³ksV™d…Šyscm›au{gsidxzpky y0 ∈ E
wŠgƒu¡Gxzt€y
H = {g ∈ G : g · y0 = y0}
• H
‚‰‡†pgdusqprŠ…Š…ƒjƒes—fg¦tdgdy G
wBgƒup…v†p…Šjƒesf‚zt‰gƒuvqprŠ…Š…ƒjƒes—fgXuˆ‡…ptf™f…drB‡{gdy¶ŽXˆkt‰gp¡T‚€™f…ƒrƒ…BugstdŽdyt„…ƒq y0
c
«”†`335Y†ù†uu (`w„ » w 3
y5
‚ l‘rxlf’ ‚ €•˜™o G
€ƒt%v q lfo E
w ‚Bƒ€n„k…f…gzeBrxl qŠ l8ofr ‚ ‡‰y•“l8…|l G/H ‚
– µ–Ÿ [ex]
¬¢BµB¥–n¦– c”šWVn¤T®–¦baB¬@VTœ––¢T™nœ–nš}Ÿ¨¦‘ŸV šV™‘˜‰™nœ–¶AšV™0®–¦‘ŸV˜¬fšXaTµ–}¤B‘œ9anšn˜™AµxŸx¥9aB¬@V0šV x
Qxœ–x¢ŸV ¤B™‘˜•xµ–nšV‘œ8˜®¤IPµ–ŸxšVGš}Ÿ¨š¸ń˜µxµ–§ ¤B Q•žWVB¦‘™BžXIGšVG¬@V‘µxŸV x
±–V– ¬fšVG®8Ÿ ¦an¦‘™‘˜P Y†™Tž} £Bµ–Ÿ3¶nšn˜•™}¤Tx˜ E1, E2
ŸV ¤n™‘˜•šV‘œ–n¦‘ T ˜®x8 †°f£´}x˜•®x™‘˜¦V
G
µx ™0šV‘œ–n¦‘ T ˜®9I
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f
ŸV ¤n™‘˜†§ ¤n™B›%˜¬–µx–´ †˜¬µ–¶x› © ™Bń˜¬fš}Ÿs´¸£‹¤
G
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¡
· !¡9£¦¥¨§¦¥©!#¥ ¡¦¥©! ¡¦¥© ¥! ¡£ Galois
u¦gƒ‡{™„†vy˜tdgp†~gprŠ‚‰uwB—p†puˆ‡g G → E
jƒ‚atfnprŠ… g → g · y0
c©• gprŠ‚‰uwB—v†suˆYg~‚‰‡†pgdu†‚€rŠ‡wBgv¡TsVy¤… E
‚‰‡†vgƒu†…Šjd…spY‚ž†pŽsy G
³BsV™f…Šypc xXx€t€y H = {g ∈ G : g · y0 = y0}
c’²3u{g†pg”gprŠ‚‰uwB…p†p‡Œd…p†vt‰gdu@—fns… ˆYt‰…Bu³k‚‰‡{g g1, g2 ∈ G
ˆkt‰… ‡{—vu… ˆkt„…Šu³k‚„‡…Dt‰…Šq E
¡Tg”rŠ™fxzrƒ‚„u
g1 · y0 = g2 · y0 ⇔ g−1
2 · g1 = y0 ⇔ g−1
2 · g1 ∈ H
c}›%gpiƒgs—fŽ‚¡TgarŠ™dx€rƒ‚‰uY†pggv†pŽswŠ…Šqp†ˆYt‰…D‡{—vu… ˆ‡nsjdrdiƒ…ƒwŠ… tfgsy H
c ¡
‰VtdˆTu§@gEgprŠ‚‰uwB—v†suˆYg G → E
‚€rƒespY‚‰u†jŠ‡{g b b wBgduR‚zrB‡gsrƒ‚‰u{wŠ—p†suˆ‡g f : G/H → E
c‘epiŠuˆkt‰g g f
‚‰‡†pgduPxž†pgsyuˆY…Šjd…Š™f‹Guˆ‡jd—Šy%gs™vuˆktf‚€™ds†† G

³BsV™dy††awBgduk…Bu G/H, E
‚„‡†pgduTuˆ‡—Šjd…Š™f‹‡…Bu‡gs™vuˆktf‚€™f…B‡ G
³BsV™f…Šuc♦
| sV™fg x€ˆkt€y G
—f™dgEjd‚€t‰gv”ŠgptvuwBe©ˆY‚¤xz†vgEˆYnp†v…sid… E
gsrƒ— —f‚€“vu{e©§Rtf—ptf‚¥iƒx€jƒ‚—ptvuV… E
‚‰‡†vgƒu¶xz†vgdy …Šjƒ…ppY‚z†vŽdym—d‚‰“fuŒ—ƒy G
³BsV™f…Šypc ¡
‰¶ˆkt‰y ϕ : E → E
xž†pgdy~gdqpt‰…Šjƒ…Š™d‹GuˆYjƒ—Šyt‰…Šq E
cm|}—ptf‚%p‡u{gmwBev¡G‚ y ∈ E
t‰g©ˆ‡gsjƒ‚‰‡{g y, ϕ(y)
x€³k…Šqv†~t‰…v†©‡{—vu…’ˆkt„gv¡T‚‰™d…ƒrŠ…BugptfŽkc$†ftv‡ˆktf™f…ƒ‹Tgmx€ˆkt‰y x, y ∈ E
rƒ…Šqmx€³k…Šqv†¥t„…v†m‡Œ—fu…©ˆYt‰gv¡G‚€™f…drŠ…BugptfŽkc¢trƒ…d¡Tx€t‰…Šqsjƒ‚i—ptvuqsrƒed™d³Y‚‰u ϕ ∈ Aut(E)
x€tdˆTuŠsVˆktf‚ ϕ(x) = y
c¶hX™f‡Œd…Šqdjd‚'tfgv† ϕ
yVy¦‚€“fŽsysc ¡
‰¶ˆYt€y z ∈ E
c|}—std‚ gprŠ—’tfgv†’jƒ‚zt‰gp”ƒgptvuwBŽ©—f™fesˆYgEtfgsy G
qprŠes™f³k‚‰u g ∈ G
tdxzt„…Šu… sVˆktf‚ z = x · g
c›%gpiƒgs—fŽ‚rƒ™fxzrŠ‚‰uŠ†pgx€³Y…ƒqdjd‚
ϕ(z) = ϕ(x · g) = (ϕx) · g = y · g.
¡
‰Vtfˆ‡u …Š™v‡{f…Šqsjƒ‚ ϕ(z) = y · g
c$ut™fxzrƒ‚„u‡†vg¤‚€“fgsˆY‹TgpiŠ‡ˆY…Šqsjƒ‚t—stfuG…~…Š™vuˆ‡jd—Šyigdqptf—Šyh‚‰‡†pgdugp†v‚‰“ded™dtdgst‰…ŠyhgprŠ—¤tfgv†‚zrBuid…spYŽ%t‰…Šq g
§G—fgpiƒgs—fŽgv† x · g = x · g ,
tf—ptf‚ y · g = y · g
§
wBeptvuYrƒ…Šq‚rƒ™f…ƒwBnprƒtd‚‰uYgprŠ—tfgv†‚qsrƒ—d¡T‚‰ˆYga—ptvukt„g x, y
x‰³k…Šqv†£t‰…p†%‡{—vu…¥ˆkt„gv¡T‚‰™d…ƒrŠ…BugptfŽkc
$ ¨'¨ (`w„ 3w ¦
ƒ„ lBr‡y p „ A
„“•e‘„‰’‰”BlBr‰l qŠ …|g‹rxl©ko ‚ €efo‘lBrxl’ ‚ €xl ©ko G
€ƒt%v q l‘’ E
‚Bƒ€B„†…zlsief‡xvh q h h l‘r‰y p„ Aut(E)
„†€€’‹…{„Ž‡‰y5
‚ p ©†ldgzh™r ‚‘ƒ„ x, y ∈ E
“•l–’ mtGl‘’k€”Bl8€
ƒp …|l gi”n„5
‚ q l•“l8…h‹”8˜kw3’ˆ“fy qt ‚ … m €n„‰oe„‰’‰”BlBrxl qŠ …gr3e8o ϕ ∈ A
w m ”–g…‹v%gi” ‚ ϕ(x) = y
t„ ‚
¡
‰¶ˆkt‰y H
qprŠ…Š…Šjded—dg‚tfgsy G
tf—ptf‚¦
N(H) = {g ∈ G : gHg−1
= H}.
‚‰‡†vgƒuƒqprŠ…ƒ…Šjƒes—fgXtdgdy G
rƒ…ŠqXrŠ‚€™vu{x€³k‚‰udtfgv† H
wBgƒud…p†p…ƒjƒesf‚zt„gduƒ…iwBgv†p…p†pu{wŠ…ƒrŠ…BugptfŽsy nor-
malizer
tfgsy H
c©‰'‡†pgduRg‘jƒ‚zpYgsidnptf‚€™fgmqprŠ…Š…ƒjƒes—fgmt‰…Šq G
rŠ…Šq©rƒ‚‰™fuŒx€³k‚‰u3tfgv† H
ˆYgp†wBgv†p…p†pu{wŠŽ%qprŠ…Š…Šjded—dgYcut™f…ƒ‹Tgv†vsVyigp† H G
td—std‚ N(H) = G
c
WEF‡%`u ¨ (xw„ ˆfw 12 gi”¸• E m €B„ko l‘rxlf’ ‚ €•˜™o G
€ƒt%v q lfoX‡k„k… m gz”¸• H
w0€n„ ‚Bƒ€n„k… lgi”n„€
5
‚ q l•“l8…h‹”8˜™o ‚ €e8o y ∈ E‚ ¡ e–” ‚ h6lBr‡y p„ „‰’‰”Bl‘rxl qŠ …|g‹rxv0€X”Bl–’ E ‚Bƒ€n„k…v…gzeBrxl qŠ h r ‚”8h‹€lBr‡y p„ N(H)/H ‚t 7
$rŠ—Et‰gmrƒgd™fgprŠev†vy ‚„‡†pgdu3t„sV™fg‘“f‚zwBep¡Tgs™f… §3—stfut„…Eˆ‡nv†p…piƒ… p−1
(x)
‚‰‡†vgƒu3xz†vgdy¥…
jƒ…ppY‚z†vŽdya—d‚‰“fuŒ—ƒy π1(X, x)
³BsV™f…ƒysc ²u{gmwBev¡T‚ y ∈ p−1
(x)
…Eˆkt‰gp¡T‚€™f…ƒrƒ…BugstdŽdy‚gdqpt‰…Šnt‰…Šq y
§V‚‰‡†vgƒuRg’qsrƒ…Š…Šjƒes—fg p∗(π1(Y, y))
tfgsy π1(X, x)
rƒ—Š™vuˆYjƒgDbdc¢

c l'qp†p‚zrdsVy¥¡Tgx€³Y…ƒqdjd‚¤t‰…”ˆYnp†p…pid… p−1
(x)
§'ˆ‡gv†Exž†pgdy …Šjƒ…ppY‚ž†pŽdy —f‚€“vu{—Šy π1(X, x)
³BsV™f…Šy §@†pg ‚„‡†pgduuˆY—Šjƒ…Š™d‹T…Šy¦jƒ‚}tfgv†iˆYqsisid…spYŽhˆYqdjsrƒid—ƒwŠy†† π1(X, x)/p∗(π1(Y, y))
UrŠgs™fgstdŽd™dgdˆYgbdcw ŠcwBgdu‡…¤gd™vu¡Tjƒ—ƒyXt€y††a‹‡nsisidy††£tdgdywBgpidqsrdtvuwBŽsytgprŠ‚‰uwB—v†suˆYgdytuˆY…Šnst‰gduYjd‚Xt‰…p†%—f‚‰‡wŠtfg‚tfgsyqsrƒ…Š…Šjded—fgsy p∗(π1(Y, y))
c‰@‡†pgƒuƒrƒ™fepp‡jdgstfuƒes“vu…irŠ™f…ŠˆY…s³YŽsy UwBgduƒgsrƒ…Š™v‡{gsy}‡ˆkyVy#E

—ptvuƒwBgpt‰gd‹‡x‰™dgdjd‚¶†pgh‹Ytfesˆ‡…Šq
jƒ‚ˆkt‰…¥¡T‚zsV™fgsjƒggprŠ—¥—ƒrŠ…ƒqa“f‚zwku†pŽsˆ‡gsjƒ‚ U”diƒxzrƒ‚trŠ—ƒ™vuˆYjƒg~bsc{w
gpidide%jdx‰ˆky —vu{gs‹T…Š™d‚€tfu
wŠs††¤ˆYqsisiƒ…pp‡uˆYjds††Bci²3u{g¤tdgp†¤ˆYqp†vx‰³k‚‰u{g¥¡Tg~—d…Šndjd‚£tfu ˆ‡³kx€ˆ‡g~jdrŠ…ƒ™f‚‰‡G†vg~x€³k‚„u g …Šjƒes—fgt€y†† covering transformations
‚ž†p—Šy$wBgpiƒqprƒtfu{wŠ…Šn³BsV™f…Šq£jƒ‚$tfgv†i—f™fesˆYg£tdgdy π1(X, x)
ˆYt‰… p−1
(x)
c
–¨² T ˜™%šWVn¤A™nœ–¶–ž}ŸV˜¡XV@Q¥‘¦‘§¨œ–Ÿ3¬fšV}¤e©¯}«Q8¬Ÿ¨¦– ž¸™T·‘ªBª8±– HUT ˜™%šWVn¤A™nœ–¶–ž}ŸV˜¡XVT¥‘¦‘§¨œ–Ÿ3¬Ÿ¨¦f±i·‘ªB%šVx¢e©¯}«±

S ¢¡¤£¦¥ § ¥©¢ ¥¡ ¥©¢ ¡¦¥ ·
s
«”`x…P5k†uu (xw¦ fw ¤3…{„p‡‰y5
‚ „‰’‰”BlBrxl qŠ …gr3e ϕ ∈ Aut(Y/X)
wz‡‰y5
‚ gzhˆr ‚Bƒl y ∈ p−1
(x)
‡‰„k… [α] ∈ π1(X, x) mtAl–’xr ‚98
ϕ(y · [α]) = (ϕy) · [α],
p h8‰„ p ˜X‡‰y5
‚ ϕ ∈ Aut(Y/X) ‚ “y8’ ‚ … m €B„•€P„‰’‰”‘lBrxl qŠ …|gr‡eR”Bl–’egv’k€feVil–’ p−1
(x)
w “„ ƒq €
€xlf€”n„‰o ”Bl p−1
(x)
gk„•€ m €B„†€ p ‚n“ …ƒe π1(X, x)
€t0v q l ‚yX“e p ‚ …“ h 8
}†p…Š™‰¡Ts††p…ƒqdjd‚£t‰… [α]
ˆY‚hwŠidedˆYg¥—f™f—Šjsy†† [˜α]
ˆkt„…v†¥³BsV™f… Y
§ jd‚£gs™f³kŽt„…~ˆ‡gsjƒ‚‰‡… y
x€tdˆTuGsVˆktf‚ p∗([˜α]) = [α]
ch|}—std‚ y · [α]
¡Tg ‚‰‡†pgdu t‰…~tdxziƒ…Šyit€y††¥—f™f—ƒjdy††
[˜α]
c | sV™fgrƒgƒ‡{™„†p…v†vt‰gsyXt‰…Šqdyh—d™f—Šjƒ…ƒqdy ϕ∗([˜α])
ˆkt„…v† Y
§k¡Gg¥x€³k…Šqv†ags™f³kŽat‰… ϕ(y)
wBgdutfxziƒ…ƒyXt‰…¤ˆYgsjƒ‚‰‡… ϕ(y · [α])
c'xXguˆY³Yns‚‰uY—ptvu
p∗(ϕ∗([˜α]) = (p ◦ ϕ)∗([˜α])
t D = p∗([˜α]) = [α].
¡
‰Vtfˆ‡u‡giwŠiƒesˆYg‚—f™d—Šjdy†† ϕ∗([˜α])
§BgprŠ…ptf‚zid…Šnp†gp†v—Š™„¡‡yVˆ‡g£tfgsytwŠidedˆYgsy [α]
wBgduBˆ‡qv†p‚zrƒsVygsrƒ—¤jd…p†vgd—vuwB—ptfgpt‰g%gv†p—Š™„¡‡yVˆYgdy §B¡Gg%rƒ™fxzrŠ‚‰uk†vg%t‰gdqptv‡{f…v†vt‰gƒuBjd‚Xt‰… [˜α]
c ¡
‰Vtfˆ‡u
y · [α] = (ϕy) · [α] = ϕ(y · [α]). ♦
WEF‡a`vu ¨ (xw¦R(`w 12 gz”¸• (Y, p)
‡‰„–‹’ˆ“•”–…s‡‰efo3t%v q lfo ”Bl–’ X
r ‚ x ∈ X
w`”xe–” ‚ h lBr‡y p „
Aut(Y/X) ‚Bƒ€n„k…3…|geBr‰l qŠ h r ‚ ”8h‹€ Aut(p−1
(x))
wd“„ ƒq €xl8€•”n„koŽ”Bl p−1
(x)
g†„†€ m €B„†€
p ‚}“ …e π1(X, x)
€ƒt%v q l ‚yX“e p ‚ …“ h 8
$† ϕ ∈ Aut(Y/X)
§}tf—ptf‚©…¬rŠ‚€™vu…Š™fuˆ‡jd—Šy ϕ|p−1(x)
‚‰‡†pgdu'xž†pgdy‘gsq
t„…ƒjƒ…Š™f‹TuˆYjƒ—Šy%tdgdy p−1
(x)
§ˆYgp† xž†pgsy%—f‚€“vu{—Šy π1(X, x)
³BsV™f…Šy §3gsrƒ—mt„…mrƒ™f…ŠgppY…Šndjd‚ž†p…¡G‚zsV™fgsjƒgkc~‰VrB‡ˆ‡gsy§3wBev¡G‚agsqpt„…ƒjƒ…Š™f‹TuˆYjƒ—Šy ϕ
§R‚€“fgd™„tfept„gdu6jƒ—v†p…p†©gsrƒ—‘t‰…v†mrƒ‚‰™fu…Š™vuˆYjƒ—t„…ƒq ϕ|p−1(x)
§Y—fgpiƒgs—fŽ‚gagsrƒ‚‰u{wŠ—p†suˆ‡g
k : ϕ → ϕ|p−1(x)
‚„‡†pgdu'bžbsc©ut™despYjƒgptvu Kerk = {ϕ ∈ Aut(Y/X) : k(ϕ) = id}
c‘$rŠ—’t‰…ErŠ—ƒ™vuˆYjƒgbdcSb¡¢ g ϕ
—f™fg~‚€id‚€np¡T‚€™fg¥ˆYt‰…v† Y
wBgduPˆYqv†p‚zrƒsVy ϕ = id
wBgƒu6g k
‚‰‡†pgƒuRbžbscit™dwŠ‚„‡G†pg—f‚‰‡Œ“„y —ptvuVg k
‚‰‡†pgƒuV‚zrB‡c”$rŠ— t‰… idŽdjdjƒgDbdcw ¢©§V…BuVgdqpt‰…Šjƒ…ƒ™f‹GuˆYjƒ…Š‡rŠ…Šq’‚zrŠeppY…Šqp†©…Bu
Aut(Y/X)
‚„‡†pgduk—sidg%ga…ƒjƒes—fg Aut(p−1
(x))
§Tgv†f†£p‡uŒg‚wBep¡T‚ y1, y2 ∈ p−1
(x)
jd‚tt‰…ŠqsyˆYt‰gv¡G‚€™f…ƒrƒ…Bugptfx€y}gdqptds††t‰y†† yi
†pg£‚„‡†pgduB‡ˆY…Bu§Š—fgpidgd—fŽ p∗(π1(Y, y1)) = p∗(π1(Y, y2))
§
qsrƒed™f³k‚‰u ϕ ∈ Aut(Y/X) : ϕ(y1) = y2
c†eheptvusrƒ…ŠqhuˆY³knd‚‰usgsrƒ—it‰…irŠ—Š™fuˆ‡jdg%bsc{wBbdc ¡
‰Vtfˆ‡u
k(Aut(Y/X)) = Aut(p−1
(x))
wBgdu k
‚‰‡†pgdu‡‚zrB‡c♦
«†…R`† ¨ (`w¦†‰w ¤f…{„Ž‡‰y5
‚ x ∈ X
‡‰„†… y ∈ p−1
(x)
w05•„ mtf•ex”‘…8
Aut(Y/X) ∼= N[p∗(π1(Y, y))]/p∗(π1(Y, y)),
e•“•l‘’ l N[p∗(π1(Y, y))] ‚‘ƒ€n„k…‡l ‡‰„•€xlf€8…s‡‰l•“•lx…h”x˜™o””8h™o ’ˆ“•lflBr‡y p„ ‰o p∗(π1(Y, y))
”Bl–’
π1(X, x)‚¦rƒ—at„…‚rŠ™f…ŠgppY…Šnsjƒ‚ž†p…£¡T‚zsV™fgsjƒg©§ƒ¡Gg‚x€³ky —ptvu Aut(Y/X) ∼= Aut(p−1
(x))
cV‰¶‹Tgs™
jƒ—Šf…v†vt‰gsytdsV™fg%t‰…a¡T‚€sV™dgdjdgmbdcw ¥©§ŠrŠ™d…ƒwBnprƒtf‚‰uBt‰…¤dgst‰…Šnsjƒ‚ž†p…Tc
WEF‡a`vu ¨ (xw¦P¦fwjyX€ (Y, p)
‡‰„•€xl8€f…—‡êfod‡‰„‘’ˆ“•”–…—‡êfo‡t0v q lfoj”Bl–’ X
w‰”–e–” ‚98
Aut(Y/X) ∼= π1(X, x)/p∗(π1(Y, y))
’‹…{„”‡‰y65
‚ x ∈ X
‡‰„†… y ∈ p−1
(x)‚ut™f…dwBnsrdtf‚‰uBedjd‚€ˆ‡g£gsrƒ—%t‰…arŠ—Š™fuˆ‡jdg¤bsc±dwp‡uŒgptv‡kgs‹T…Šn p∗(π1(Y, y)) ¡ π1(X, x)
§Š¡Ggx‰³By —stfu N[p∗(π1(Y, y))] = π1(X, x)
c
«†…R`† ¨ (xw¦ ¯ wŽy€ l (Y, p) ‚‘ƒ€n„k…‰‡‰„5klsi…—‡êfo‡k„–‹’ˆ“†”‘…s‡ê8oGt0v q lfo ”Bl–’ X
w 5†„ mtf•
Aut(Y/X) ∼= π1(X, x)
– S ¥‘¦‘§¨œ–Ÿ functorial
˜žn˜¶nšXVBš¸Ÿs›uš ™¤0Ÿ¨œx™ T ¶xµ–Ÿ ¤ ™¤G–µx–µx–´ †˜¬µ–£‹¤R©ª}«Q¬Ÿ¨¦f±|¹¯8±

fv · !¡9£¦¥¨§¦¥©!#¥ ¡¦¥©! ¡¦¥© ¥! ¡£ Galois
(xw ( ! ¢ ) 3IHPIR87'H¶Q ) 1P2†45 87'H¶Q¡ Ea`PH¶ 7@V¢ %`3H¶ « u1RQ7'
¡
‰¶ˆYt€y (Y, p)
wBgpidqsrdtvuwB—Šy³BsV™f…ƒy£t‰…Šq X
c}‰VrŠ‚‰u{—fŽ¥g p
‚‰‡†pgduGjŠ‡{g¥gp†p…Šu{wƒtfŽ¥gsrƒ‚‰u{wŠ—p†suˆ‡g©§
… X
x€³Y‚‰u'tdgp†”t‰…ƒrƒ…sid…ppT‡{g rƒgsiƒ‡wB… rƒ…ŠqD‚zrŠepp‡‚zt‰gdu'gsrƒ—¬tdgp† p
c ¡
‰Vtfˆ‡u¦jsrŠ…Š™f…ƒndjd‚~†pgrŠes™f…Šqsjƒ‚it‰…p† X
gsrƒ—mt„…v† Y
t‰gdqptv‡{f…v†vt‰gsy‚ˆYqspkwB‚zwB™fuŒjdxž†pg~ˆYgdjd‚‰‡Œg²3u{g wBev¡T‚ x ∈ X
§
—sidgt‰ghˆ‡gsjƒ‚‰‡{gt‰…ŠqhˆYqp†p—pid…Šq p−1
(x)
§ƒrŠ™fxzrƒ‚„up†pght„gsqptvuˆkt‰…Šnp†tˆY‚}xž†pghˆ‡gsjƒ‚‰‡…‡c†• …Šjƒes—fg
Aut(Y/X)
§3jƒ‚zt‰gv¡Gxztf‚‰u6—sidg t‰gmˆYgsjƒ‚‰‡{g~t‰…ŠqmˆYqp†p—pid…Šq p−1
(x)
§jƒ‚zt‰gs“fn t‰…Šqsysca²3‚ž†suwBe—f‚ž†iuˆY³knd‚‰us—stfu Y/Aut(Y/X) ∼= X
c†©rŠ…ƒ™f‚‰‡s†pgiqsrƒed™d³Y…ƒqp† y1, y2 ∈ p−1
(x)
wBgdu ϕ ∈
Aut(Y/X)
xztfˆ‡udsVˆktf‚ ϕ(y1) = y2
cR›%gpidgd—fŽt…Bu Aut(Y/X)
†vgjƒgv†—d™f…Šqv†Xjd‚zt„gv”ƒgstfu{wŠeˆYt‰… p−1
(x)
c
$ ¨'¨ (`w¦ ° w y€ (Y, p)
‡‰„‘’ˆ“•”–…—‡êfo t%v q lfo ”Bl–’ X
w‘h£lBr‡y p„ „‰’‰”‘lBrxl qŠ …|gr`v0€
Aut(Y/X)
pVq „ r ‚ ”B„ v„”–…—‡‰y‘gz”Bl p−1
(x)
wPr ‚ x ∈ X
„†€€6l (Y, p) ‚Bƒ€B„†…3‡‰„•€xlf€8…s‡‰efo
‡‰„‘’ˆ“•”–…—‡êfo3t%v q lfo ‚y“•e p ‚ …“ h 8
• …Šjƒes—fg Aut(Y/X)
—d™fg jƒ‚zt‰gp”ƒgptvuwBe ˆYt‰…v† p−1
(x)
gp† p‡u{g wBev¡G‚
y1, y2 ∈ p−1
(x)
qprŠes™f³k‚‰u ϕ
xztfˆ‡ufsVˆYtd‚ ϕ(y1) = y2
c¦qptf— §pgprŠ—¦t„…trŠ—Š™vuˆ‡jdg£bsc{wBb§dgdqptf—¡GgˆYqsjs”ƒgƒ‡†p‚‰uTgv†f† p∗(π1(Y, y1)) = p∗(π1(Y, y2))
c}$rŠ—¡G‚zsV™fgsjƒg©bsc·©§Y…BuTqprŠ…Š…Šjded—d‚‰y
p∗(π1(Y, y))
p‡u{g y ∈ p−1
(x)
gsrƒ…std‚ziƒ…Šnv†¥jƒ‡Œg¤wŠidedˆYg~ˆYqddqsp‡‡{gdy‚gsrƒ—©qsrƒ…Š…Šjded—f‚€y£t„…ƒq
π1(X, x)
c ¡
‰VtdˆTu p∗(π1(Y, y1)) = [α] ∗ p∗(π1(Y, y2)) ∗ [¯α]
ci$rŠ—mtfgv†¥wBgv†p…v†suwB—stdgst‰gt‰…Šq%wBgpiƒqprƒtfuwB…Šni³BsV™f…ŠqxzrŠ‚zt‰gƒuk—ptvu p∗(π1(Y, y1)) = p∗(π1(Y, y2))
c♦
‰@‡{—fgdjd‚—ptvu‡gv†‚t„… (Y, p)
‚„‡†pgduYwŠgp†p…v†suwB—Šy¦wBgpiƒqprƒtfu{wŠ—Šyt³BsV™f…ƒyt„…ƒq X
§Btf—ptf‚¦
Y/Aut(Y/X) ∼= X.
tyh—d…Šndjd‚tt„sV™fg%t‰… gv†vtf‡ˆYtd™f…Š‹‡…¥tdgdytrŠ™f—pt‰gdˆYgdy‚bdcSbfµ
«”`x…P5k†uu (`w¦ ¥3w 12 gz”}• e–”–…l (Y, p) ‚Bƒ€n„k…‹‡‰„‘‹’ˆ“†”‘…s‡êfoxt%v q lfoe”Bl–’ X ‚ ¡ e–” ‚ hŽl‘r‰y p„
Aut(Y/X)
w pVq „ evenly
gi”Bl8€ Y ‚ 2 “ ƒgzh‹oe„•€h Aut(Y/X)
pVq „Žr ‚ ”n„v„”‘…s‡‰yXg ‚ ‡‰y5
‚Š ©Bkild”xh‹o p
wx”–e–” ‚ ”Bld‡‰y–‹’xr‡rv„ ‚Bƒ€B„†… m €B„X‡‰„†€xl8€8…s‡ê G
€s‡‰y‘’xr‡rv„jr ‚ Aut(Y/X) = G‚y“•e p ‚ …“ h 8
xXg —d‚„‡{“f…ƒqdjd‚©—stfu$gD—f™dedˆYg¬‚‰‡†pgdu even
c ¡
‰¶ˆkt€y y ∈ Y
wBgdu N
jŠu{gpY‚„ut„…v†su{e t‰…Šq p(y)
§3g~rŠ…B‡{g ‚„‡†pgdu3…ƒjƒgpiƒe¥wBgpiƒqsjƒjdxz†vg~gprŠ—mtfgv† p
cutgd‡{™„†vy xz†vgm‹Tnpidid…tfgsy p
§ V
jƒ‚ y ∈ V
x€tdˆTuPsVˆYtd‚g p|V V → N
†pg©‚‰‡†pgduxž†pgdy%…Šjƒ…Bu…Šjƒ…Š™d‹GuˆYjƒ—Šypc $†
ϕ = ϕ ∈ Aut(Y/X)
§ktf—ptf‚ ϕ(V )
wBgdu ϕ (V )
§‡¡TgarŠ™fxzrƒ‚„uB†pg¥‚‰‡†vgƒuT“dxz†vg¨§Y—vu{gs‹T…Š™f‚ztfu{wŠeg ϕ−1
◦ ϕ
¡Tg¥x‰³k‚‰uGxž†pg¥ˆYt‰gv¡G‚€™f— ˆ‡gsjƒ‚‰‡… ˆYt‰… V
§‡rŠ™feppYjƒg¥est‰…ƒrƒ… gprŠ—¥rŠ—ƒ™vuˆYjƒg©bdcSb¡¢kc¡
‰Vtfˆ‡u ϕ(V ) ∩ ϕ (V ) = ϕ · V ∩ ϕ · V = ∅
c¦| sV™fg¤gsrƒ— t‰… rƒ™f…ŠgppY…Šndjd‚ž†p… idŽdjdjƒggv†g‚…ƒjƒes—fg Aut(Y/X)
§B—f™dg%jƒ‚zt‰gv”ŠgptvuwBehˆkt‰… p−1
(x)
§Y… Y
‚‰‡†pgduŠwBgp†v…p†suwB—ƒy¦wBgpidqsrdtvuwB—Šy³BsV™f…Šypc ¡
t™fg Y/Aut(Y/X) ∼= X
wBgduYrƒ™f…Š‹‡gp†vsVy G = Aut(Y/X)
c♦
utgs™fgptfgs™f…Šnsjƒ‚©—ptvu$gp†”g —f™fesˆYg¬‚‰‡†pgdu¦jd‚€t‰gv”ŠgptvuwBŽ©§'¡Tg ‚‰‡†pgdu}wŠgƒu rBuˆYtdŽ¨§¦—fgpiƒgs—fŽ”t‰…
ϕ ∈ Aut(Y/X)
§'jƒ‚ ϕ(y) = y
‚‰‡†pgƒu¶jƒ…v†pgs—vuwB—Tc ¦rƒ— t‰… ¡T‚€sV™dgdjdg bdc±d± wŠgƒuVtfgv†rŠ™f—pt‰gdˆYg bsc±s·%rƒgƒ‡{™„†p…ƒqdjd‚¦t‰…rŠgs™fgpwBest€yDrŠ—ƒ™vuˆYjƒg
«†…R`† ¨ (xw¦ » wjyX€ (Y, p) ‚Bƒ€B„†… m €n„‰od‡k„†€xl8€f…—‡êfoe‡k„–‹’ˆ“†”‘…s‡ê8o‡t%v q lfoj”‘l‘’ X
w‰”xe–” ‚ l
Y ‚Bƒ€B„†… m €B„‡‰„†€xl8€8…s‡ê G
€V‡ky–‹’xr‡rv„‡k„k…
G = Aut(Y/X) ∼= π1(X, x)/p∗(π1(Y, y))
’‹…{„†‡‰y5
‚ x ∈ X
‡‰„†… y ∈ p−1
(x)‚ yX€ l Y ‚‘ƒ€n„k…%„•“}iy g’†€ ‚ ‡™”‘…s‡êfoet0v q lfo ”–ex” ‚
π1(X) ∼= G = Aut(Y/X)‚‰@‡{—fgdjd‚h—stfuTxž†pgsywBgsidqprƒtvuwB—ƒyX³BsV™f…Šy (Y, p)
t‰…Šq X
wBgp¡T…Š™v‡{f‚zt‰gdukrdiƒŽs™dyVyhgprŠ—¥tfgv†wŠidedˆYg~ˆYqddqsp‡‡{gdy£tfgsy%qprŠ…Š…ƒjƒes—fgdy p∗(Y, y)
t‰…Šq π1(X, x)
c|}…‘‚‰™„sRtfgsjƒg~rƒ…Šq rƒ™f…ƒwBn
rƒtd‚„u ‚‰‡†pgdu6—ptvutgp† X
‚‰‡†pgdu6xž†pgsy‚t‰…ƒrƒ…sid…ppTuwB—ƒyh³ksV™d…Šy‚wBgdu gp†¤jdgdy‚—v‡†p‚zt‰gƒu jŠ‡{g¤wŠiƒesˆYgˆ‡qsfqpp‡‡Œgsy£gsrƒ—©qsrƒ…Š…Šjƒes—f‚€yit„…ƒq π1(X, x)
§qprŠes™f³k‚‰u wŠgsidqsrdtvuwB—Šyi³BsV™f…Šy (Y, p)
t‰…Šq X
xztfˆ‡uYsVˆktf‚g p∗(Y, y)
†vgagp†vŽswB‚‰uYˆ‡‚tgdqptfŽv†£tfgv†‚wƒiƒesˆ‡g‚ˆ‡qsfqpp‡‡Œgsy¦E

S ¢¡¤£¦¥ § ¥©¢ ¥¡ ¥©¢ ¡¦¥ ·
„˜
WEF‡a`vu ¨ (`w¦ 3w 12 gi”¸• X
”Bl•“•lsil’‹…—‡êfo‰t0v q lfonw‡“l–’ mt ‚ …‡‰„5klsz…s‡êR‡‰„–‹’ˆ“•”–…s‡‰eft0v3€
q l ‚ ¡ ex” ‚ ’™…„ ‡ky65
‚ ‡xzy‰gzhgv’¡B’’ ƒ„‰oA„•“ee’ˆ“•lflBr‡y p ‚ oT”8h™o π1(X, x)
wx’ˆ“y qt ‚ …‰‡‰„‘’ˆ“•”–…€
‡êfoft%v q lfo (Y, p)
”‘l‘’ X
w m ”xg‹…™v%gi” ‚ h p∗(π1(Y, y))
€n„„†€˜™‡ ‚ …ig ‚ „‰’‰”8˜‹€ ”8h‹€d‡xzy‰gzh
g’¡B’’ ƒ„‰o ‚
yX“e p ‚ …“ h 8
¡
‰@ˆkt€y ( ˜X, q)
…mwBgp¡T…piŠuwB—Šy£wBgpiƒqprƒtfu{wŠ—Šyi³ksV™d…Šy%t‰…Šq X
ca• π1(X, x)
—f™fg‘jƒ‚zt‰gp”ƒgptvuwBe¤ˆYt‰…’ˆ‡nv†p…piƒ… q−1
(x)
gprŠ—‘—f‚€“vu{e©§3wŠgƒu6wBgp¡‡sVy… Y
‚„‡†pgdugprƒide~ˆYqp†v‚
wŠtvuwB—Šy$—f™dg‚‚zid‚‰nv¡T‚‰™dgYc3‰¶rŠ‡ˆYgdy}wBgduŠgi…Šjƒes—fgigdqpt‰…Šjƒ…ƒ™f‹GuˆYjds†† Aut( ˜X/X) ∼= π1(X, x)
—f™fg©jƒ‚zt‰gv”ŠgptvuwBe$UwBep¡T‚‚wŠgp¡T…siƒuwB—~wBepidqdjdjƒg~‚‰‡†pgdu6wŠgƒu wBgv†p…p†pu{wŠ—
gprŠ—©gs™vuˆktf‚€™fe©§3ˆkt‰…ˆ‡nv†p…piƒ… q−1
(x)
cV›%uŒgpidx€pkyExž†pgˆYgdjd‚„‡… ˜x ∈ q−1
(x)
wBgdusqprŠ…ƒ…Šjƒes—fg G
rŠ…ƒqgp†vŽswB‚‰upˆYtdgp†wŠiƒesˆYg£ˆYqsfqsp‡‡{gsy$rƒ…Šqijƒgsy$—v‡†p‚zt„gduc ¡
‰¶ˆkt‰y H
†pg£‚‰‡†pgƒuŠqprŠ…ƒ…Šjƒes—fgt€y††£gdqpt‰…Šjƒ…ƒ™f‹GuˆYjds††
Aut( ˜X/X)
rŠ…Šq©…Š™f‡Œd‚€t‰gdu3yVy¥‚€“fŽdy ϕ ∈ H
gv†f†mqsrƒed™f³k‚‰uˆkt‰…Bu³k‚‰‡… [α] ∈ G
tfxzt‰…Bu…sVˆYtd‚ ϕ(˜x) = ˜x·[α] ∈ q−1
(x)
c@l@t‰…%—f‚€“v‡Bjƒxzid…Šy}tfgsy¦uˆ‡—ptfgpt‰gdy}x€³k…Šqsjƒ‚'tdgp†h—f™fesˆ‡ght‰…Šqgdqpt‰…Šjƒ…Š™d‹GuˆYjƒ…Šn ϕ
ˆktfgv† q−1
(x)
‚ž†vs®ˆYt‰…‘gs™vuˆktf‚€™f—‘jƒxzid…Šy‚tdgdyuˆ‡—ptfgpt‰gdy£tfgv†~—d™fedˆYgt„…ƒq©ˆYt‰…Bu³k‚‰‡…Šq¤tfgsy…Šjƒes—fgsy G
ˆkt‰…p† q−1
(x)
cm‰@‡{—fgdjd‚a—ptvu $…Š™vuˆ‡jd—Šymbdcwp´ G ∼= H
jƒx€ˆYy tfgsygp†ftvuˆkt‰…Bu³Y‡{gdy ϕ ↔ [α]
§Tgv†f† ϕ(˜x) = ˜x · [α]
c‰¶rƒ‚‰uŒ—dŽ H ≤ Aut( ˜X/X)
§kg H
—f™dg evenly
ˆkt‰…p† ˜X
c ¡
‰¶ˆYt€y Y
†pga—fgpiss††p‚‰ukt‰…v†%³BsV™f…rŠgpiŠ‡wB… ˜X/H
§ r : ˜X → Y
§G†pg ‚‰‡†vgƒu6g~‹‡qdˆ‡uwBŽ¤rƒ™f…d”ƒ…sidŽwBgdu p : Y → X
†pg~‚‰‡†pgduPggsrƒ‚„uwB—v†suˆYgrŠ…ƒqh‚€rƒespY‚‰udg q : ˜X → X
c ¡
‰¶³Y…ƒqdjd‚ t„…‚jƒ‚zt‰gp¡T‚ztvuwB—i—vu{epp‡™dgdjdjƒghxztfˆ‡uŠsVˆYtd‚
p ◦ r = q

( ˜X, ˜x)
r
−→ (Y, y)
q p
(X, x)
¡
htrŠ…Šq¤… ( ˜X, q)
wBgpiƒqprƒtfuwB—Šy³BsV™f…Šy£t‰…Šq X
gprŠ—mqprŠ—d¡T‚€ˆ‡g
§G… ( ˜X, r)
‚‰‡†vgƒuGxž†pgsy H

wBgsidqprƒtvuwB—ƒy‚³BsV™f…Šy‚t„…ƒq Y
gsrƒ—miƒŽsjƒjdgEbsc‘bdw
wBgdu (Y, p)
wBgpiƒqprƒtfu{wŠ—Šyi³ksV™d…Šyat‰…Šq X
gsrƒ—mt„…©idŽdjdjƒg’bdcws±¥p‡uŒgptv‡6gprŠ…ptf‚ziƒ‚‰‡ xž†pgv†~…ƒjƒ…Šjd…Š™f‹Guˆ‡jd—‘wŠgsidqsrdtvuwŠs††¥³BsV™dy††Šct™„wB‚‰‡†pgE—d‚„‡{“dy —ptvu G ∼= p∗(π1(Y, y))
c ¡
‰¶³By —ptvu p : Y = ˜X/H → X
‚‰‡†vgƒuRjŠ‡{g‘wBgpiƒq
rƒtvuwBŽ~gsrƒ‚„uwB—v†suˆYg©§6x€tdˆTu p∗(π1(Y, y)) ∼= π1(Y, y)
wBgdu3gprŠ—mt‰…‘rŠ—Š™vuˆ‡jdgEbsc± ¥¡Gg~x‰³By
π1(Y, y) ∼= H ∼= G
§‡xztdˆTu G ∼= p∗(π1(Y, y))
c ♦
§ Ù ¨ …RA (xw¦Pˆfw 12 €B„kovt%v q lfo X
l8€xlBr‡y¡
‚ ”n„†…•”Bl•“†…s‡kyA„•“}iyGgv’k€ ‚ ‡ˆ”–…s‡‰efo‡„†€T‡‰y5
‚ ’ ‚ …”Bl–€
€f…{y ‚ €fefoPgihˆr ‚Bƒl‘’”Bl–’w‰“ ‚ q …mt ‚ …ˆrx…{„’ ‚ …”Blf€8…{y”‘l‘’gihˆr ‚Bƒl–’“l–’ ‚Bƒ€n„k…™„“nzy g’†€ ‚ ‡™”‘…s‡™˜ ‚
§ Ù ¨ …RA (`w¯ 3w 12 €B„ko3t%v q lfo X
l8€xlBr‡y
‚ ”n„k… semilocally
„“nzyjg’†€ ‚ ‡™”–…—‡êfoG„†€d‡‰y65
‚gihˆr ‚Bƒlp”‘l‘’ mt ‚ …i’ ‚ …”Bl8€8…yj” m ”‘lx…„Pv%gi” ‚ ‡‰y5
‚ ‡x ‚ …gi”–ePrxl8€xl•“yf”–…™g ‚ „‰’‰”8˜‹€p”xh‹€ ’ ‚ …”Bl8€8…y
‚‘ƒ€n„k…zlBr‰l–”Bl•“†…—‡êer ‚ ”Bl” ‚ ”q …{r‡r m €xl ‚¡
g™l p ©†€n„xrv„„†€j‡ky65
‚ gihˆr ‚Bƒl x ∈ X mt ‚ … ’ ‚ …”Bl8€f…{y
V m ”xg‹…0v%gz” ‚ l ‚ “f„8’ˆv‡r ‚ €xlfo‘„•“e ”Bl8€ ‚ ’‡– ‚ …|g‹r3e lBrxlBr‰l q‹Š …|g‹r3efo i∗ : π1(V, x) →
π1(X, x) ‚Bƒ€B„†…Tl rfh p ‚ €f…—‡êfonw p h8k„ p ˜ „•“ ‚ …—‡ˆl8€ ƒ
‚ …T‡‰y5
‚ gi”Bl8…t ‚‘ƒl ”xh™o π1(V, x)
gi”Bl
”B„‰’‰”Bl–”–…—‡ê ‡‰„†…vgv’k€ ‚ “xv%odh i ‚Bƒ€B„†…zr3h l–’kg‹…|v p h‹o t GŽgv’k€8y q ”8h™gzh A
null-homotopicC‚
‰'‡†pgdu}rŠ™f…Š‹‡gp†vx‰yE—stfu¦gv† xž†pgsyE³BsV™f…Šy ‚„‡†pgdu t‰…ƒrBuwBe gsrdiƒeDˆYqp†v‚€wƒtvuwB—Šy §$tf—ptf‚©¡Gg‚„‡†pgduPwŠgƒu semilocally
gprƒide~ˆYqv†p‚zwŠtvuwB—Šypc‚‘rƒ…Š™f…Šnsjƒ‚£jƒepiƒuˆYt‰g†pgm—f‚‰‡{“f…Šqsjƒ‚£—ptvu3xž†pgsyˆ‡qv†p‚zwŠtvuwB—ƒy¶wBgdudt‰…ƒrBuwBe}—d™f…Šjƒ…ƒˆ‡qv†p‚zwŠtvuwB—ƒy@³BsV™f…Šy¶x€³k‚„ufwBgv¡T…siƒuwB—¦wŠgsidqsrdtvuwB—¦³BsV™f…gv†f†‚„‡†pgdu semilocally
gprƒideˆ‡qv†p‚zwŠtvuwB—ƒy t ¢ c¦²u{gt‰g¥rƒgd™dgswBept€y §Yqsrƒ…d¡Gxzt‰…Šqsjƒ‚—ptvuG…~³ksV™d…Šy
X
‚‰‡†pgƒuVˆYqp†p‚zwŠtfuwB—Šy §¶t‰…ƒrŠuwBeE—d™f…Šjƒ…ƒˆ‡qv†p‚zwŠtvuwB—ƒy wBgdu semilocally
gprƒideEˆYqp†v‚€wƒtvuwB—Šy §
x€tdˆTuYsVˆYtd‚t†pgax€³k‚‰uYwBgv¡G…piƒu{wŠ—awBgsidqprƒtvuwB—%³ksV™d…Tc¢ tfesˆ‡gsjƒ‚¶ˆYt‰…v†¦ˆYwŠ…ƒrŠ—hjdgdy §vrŠ…Šqt‚‰‡†vgƒud†pg$”Š™f…ƒndjd‚¶tdgp†¦gv†vtvuˆYt‰…Bu³Y‡{g}rŠ…Šq¦qprŠes™f³k‚„uvjd‚€t‰gs“fnt‰y††EqprŠ…Š…ƒjƒes—dy††©tfgsy¤¡T‚€jƒ‚ziƒu{sV—d…Šqdy¤…Šjded—fgsy¤‚z†v—Šy~³BsV™f…Šq’wBgduRt€y††‘wBgpidqdjdjƒept‰y††mt„…ƒq
– `¥‘¦‘§¨œ–ŸA©ª¸«‹¬Ÿ¨¦– ž}™TºB¯8±– ¥‘¦‘§¨œ–ŸA©¹¨«‹¬Ÿ¨¦– ž}™R¹WRPR8±

„ · !¡9£¦¥¨§¦¥©!#¥ ¡¦¥©! ¡¦¥© ¥! ¡£ Galois
³BsV™f…Šq¥gdqpt‰…Šnkc}|}… rƒgd™fgpwBept€y gprŠ…ptfxziƒ‚€ˆYjƒg%¡Tg¥‚„‡†pgduGjŠu{g¥‹Tqsˆ‡u{wŠŽˆ‡qv†pxzrŠ‚‰u{gatfgsyijƒx€³k™vutdsV™fgarƒ…Š™f‚‰‡{gdytjƒgsy
WEF‡%ù ¨ (xw¯ (`w Ar„ C
¤3…{„”‡‰y65
‚ ’ˆ“lflBr‡y p „ H
”Bl–’ π1(X, x)
’ˆ“y qt ‚ … m €B„”gv’†€ ‚ ‡™”–…—‡ê
‡‰y‘’xr‡rv„ 8 pH : (YH , yH ) → (X, x)
wer ‚ yH ∈ p−1
(x)
w m ”–g…fv%gi” ‚ h ‚ …s‡ê8€B„†”‘l‘’
π1(YH , yH)
gi”xh‹€ π1(X, x)
wRr m gk• ”8h™o pH∗
€n„ ‚Bƒ€n„k…%h ’ˆ“•l8l‘r‰y p„ H ‚ 3
y5
‚ y‘kil
” m ”Bl8…|l ‡‰y‘’xr‡ri„ A •%oX“ q lfod”xh‹€ ‚ “†…vlf’z˜ vykgih™o C ‚Bƒ€B„†…i…|geBr‰l q‹Š l r ‚ „‰’‰”–e ‚A
C
y€ K ‚Bƒ€B„†…0r ƒ„†y‘†h ’ˆ“lflBr‡y p „ ”Bl–’ π1(X, x)
w”“l–’ “ ‚ q …mt ‚ …G”Bl H
w’ˆ“y qt ‚ …
r‰l8€n„ p …—‡™˜ gv’†€ ‚tP˜™oe„•“ ‚ …s‡ê8€8…|gzh pH,K : (YH , yH) → (YK, yK)
“•l‘’ ‚Bƒ€B„†…zg’xr v„”x˜ r ‚”–…|od“ q l xls m opgz”Blf€ X ‚ yX’‰”8˜ ‚‘ƒ€n„k…z‡‰„‘’ˆ“•”–…—‡™˜Ž„•“ ‚ …—‡ê8€f…gih ‡‰„k…„†€ H ¡ K
w‰”xe–” ‚p‚Bƒ€B„†…
m €B„ G
€s‡‰y–‹’xr‡rv„r ‚ G = K/H ‚
y“•e p ‚ …“ h 8
8g
¡
‰¶ˆkt€y ( ˜X, u)
†pgm‚‰‡†pgduP…©wBgv¡G…piƒu{wŠ—Šy£wŠgsidqsrdtvuwB—Šyi³BsV™f…Šy‚t„…ƒq X
ct‹‡…Šn H ≤ π1(X, x)
§6gsrƒ—~t‰… ¡T‚€sV™dgdjdg’bsc± ¢ qprŠes™f³k‚‰uTwBgpiƒqprƒtfu{wŠ—Šyh³BsV™f…Šy (YH , pH )
t‰…Šq X
§$x€tdˆTu'sVˆktf‚‘g pH∗ (π1(YH , yH)) = H
c ¡
‰¶ˆkt€y pH : (YH , yH) → (X, x)
jŠ‡{g©epididg~wBgpiƒqprƒtfuwBŽ~gprŠ‚‰uwB—v†suˆYgmjƒ‚ p(yH ) = p(yK) = x
wBgdu pH∗
(π1(YH , yH )) =
H
c $rŠ—¬t‰…DrŠ—Š™fuˆ‡jdg bdcwkb ¡GgDx€³By —stfu'…Bu —fns… gdqpt‰…B‡¶wBgpidqsrdtvuwB…B‡V³BsV™f…Bu@¡Tg ‚‰‡†pgduuˆY—Šjƒ…Š™d‹T…BuRp‡uŒgptv‡ pH∗ (π1(YH , yH )) = pH∗
(π1(YH , yH ))
X… p∗
‚‰‡†pgdu¶gv†p‚€“fes™dtfgpt‰…Šygsrƒ—tfgv†%‚zrŠu{id…pp‡Žh”ƒedˆYgdypc
cU”
¡
hXjd…Bu{g%qprŠes™f³k‚„u (YK , pK)
wBgsidqprƒtvuwB—ƒyt³BsV™f…Šyt‰…Šq X
§kxztdˆTu‡sVˆktf‚
pK∗ (π1(YK , yK)) = K
c ¡
‰Vtfˆ‡uY¡Tgax€³Y…ƒqdjd‚tt‰…rŠgs™fgswŠest€y jƒ‚zt‰gp¡T‚ztvuwB——vu{eppY™fgdjdjƒg
( ˜X, ˜x)
rK rH
YK ↓ YH
pK pH
(X, x)
$†‚tdsV™dg H ⊂ K
qprŠes™f³k‚‰u pH,K
xztfˆ‡u‡sVˆktf‚Xt‰…¥rŠgs™fgswŠest€y¬—vu{espY™fgsjƒjdg‚†pg‚‰‡†pgƒuYjƒ‚zt‰g
¡G‚ztfu{wŠ—!
(YH , yH )
pH,K
−→ (YK, yK)
pH pK
(X, x)
ut™despYjƒgptvu§PgprŠ—©t‰…©¡T‚zsV™fgsjƒg bsc¡ ¥¡Tgmx€³By —ptvuPqsrƒed™f³k‚‰u6jƒ…v†pgd—fu{wŠŽ~gv†p—Š™„¡‡yVˆ‡gmtdgdy pH
jƒ‚ pK ◦ pH,K = pH
gp†f† H = pH∗ (π1(YH , yH)) ⊂ pK∗ (π1(YK, yK)) = K
c| sV™fggd‹‡…Šnag pH,K
‚‰‡†pgdu‡xž†pgsyh…Šjƒ…Šjd…Š™f‹Tuˆ‡jd—ŠywBgpiƒqprƒtfuwŠs††‚gsrƒ‚„uwB…v†s‡ˆY‚zy†† §kˆYndjd‹‡y††pg¥jƒ‚tt„…iƒŽsjƒjdg bsc{wp±¨§Š¡Tga‚‰‡†pgdu‡jŠ‡{g%wŠgsidqsrdtvuwBŽ£gsrƒ‚‰u{wŠ—p†suˆ‡gkc$† H = pH∗ (π1(YH , yH)) ¡ pK∗ (π1(YK, yK)) = K
td—std‚tgprŠ—£t„…‚rŠ™f—pt‰gdˆYg¤bsc±s·hwBgdut‰…¡T‚zsV™fgsjƒg~bsc±s±¨§k…¥³BsV™f…Šy (YH , pH,K)
§B¡Gg‚‰‡†pgdu‡xž†pgsy G
wŠgsidqsrdtvuwB—Šy¦³BsV™f…Šyhjd‚
G = π1(YK , yK)/pH,K∗ (π1(YH , yH )).
¡
hXjdyVy π1(YK , yK) ∼= pK∗ (π1(YK , yK)) = K
§Rp‡u{gptv‡Rg pK∗
‚‰‡†vgƒuRjƒ…v†p…Šjd…Š™f‹Tuˆ‡jd—Šypc¡
hXjƒ…ŠuŒg pH,K∗ (π1(YH , yH )) ∼= π1(YH , yH ) ∼= H
c¡
‰Vtfˆ‡u G ∼= K/H
c ♦
«†…R`† ¨ (xw¯ ‡w Œ “fy qt ‚ …†r ƒ„d~8€~R‡k„k… ‚ “ ƒ „†€”–…|gz”Bl8…t ƒ„er ‚ ”n„ “ ©Pes•%€R”¸•0€d’ˆ“lflBr‡y p• 0€
H
”8h™o π1(X, x)
‡‰„†… eV‹•0€Ž”¸•0€‡‰„–‹’ˆ“•”–…s‡•v0€‡xzy‰g ‚ •%€xt ¡ [pH : (YH , yH) → (X, x)]‚¢ h8‰„ p ˜ 8
H = pH∗ (π1(YH , yH )) ←→ [pH : (YH , yH ) → (X, x)].
– ¢ žWVB¦‘™BžXIdš ™¤p®x™n¦‘¢nœ‘šn˜®8£™¤d°8£z´‹¤RšVx¢ X
œ––¢e™}¤nš}˜¬fšV8˜°f £}¤d¬fšWVn¤d¢nœ––xµ9aBž}™ H modulo
© ˜¬¡
µx–´ †˜¬µ–8 ®8™n¦‘¢nœ‘š}˜®x£‹¤T°f£´ ™¤ ¨±

v ¡£ Galois ·
“
xXgax€³Y…ƒqdjd‚¦tdgp†%‚€“fŽdyXgp†vtfuˆkt„…Šu³Y‡Œg
( ˜X, ˜x) ↔ {e} = Aut( ˜X/ ˜X)
↓ ∩ ∧
(YH , yH ) ↔ H = Aut( ˜X/H)
↓ ∩ ∧
(YK, yK) ↔ K = Aut( ˜X/K)
↓ ∩ ∧
(X, x) ↔ G = Aut( ˜X/X)
$†ig H
‚‰‡†pgduƒwBgp†v…p†suwBŽhqprŠ…ƒ…Šjƒes—fgtfgsy K
§stf—ptf‚$…£wBgsidqprƒtvuwB—ƒy ³BsV™f…Šy (YH , pH,K)
¡Tg‚„‡†pgdu‡xž†pg K/H
wBepiƒqsjƒjdgYcP|}—ptf‚t¡Tgax€³k…Šqsjƒ‚X—ptvu
K/H ∼= Aut( ˜X/YK)/Aut( ˜X/YH) ∼= Aut(YH /YK).
ehep¡T‚~wBepiƒqsjƒjdg YH → X
§'rŠ…Šq gv†vtfuˆYt‰…Bu³Y‚‰‡¶ˆYtdgp† H
§ jdrŠ…ƒ™f‚‰‡¶†pg t‰gdqptvuˆYtd‚„‡¶jƒ‚ t„…
˜X/H → X
§†jd‚ H
†pg‘‚‰‡†vgƒu†jƒuŒg©qprŠ…ƒ…Šjƒes—fg~t€y†† Aut( ˜X/X) ∼= π1(X, x)
§†rŠ…ƒq©—f™fgˆYt‰…v† ˜X
cehgv†p…v†suwBe wBgpiƒnsjƒjdgst‰g't‰…Šq}³BsV™f…ƒq X
gv†vtfuˆYt‰…Bu³Y…ƒnp†'ˆ‡‚†wŠgp†p…v†suwBx€yRqprŠ…Š…Šjded—d‚‰y
H
c ¡
‰Vtfˆ‡uwBev¡G‚%wŠgp†p…v†suwB—©wBepiƒqsjƒjdg p : Y → X
§†x€³k‚‰u3tfgv†mjd…Š™f‹‡Ž ˜X/H → X
wŠgƒu‚„‡†pgdu G
„wBepiƒqsjƒjdg'UrŠ™f—pt‰gdˆYg~bsc± Bc
§Bjd‚
π1(X, x)/H ∼= Aut(Y/X) ∼= G.
'iƒxzrƒ…Šqdjd‚a—ptvu†…Bu†jƒu{wŠ™f—std‚€™f‚€y¤qprŠ…ƒ…Šjƒes—f‚€yatfgsy π1(X, x)
wBgdu†ˆYqp†p‚zrdsVy¤…Šu†jŠuwB™f—ptf‚€™f‚€yqsrƒ…Š…Šjƒes—f‚€yttdgdy Aut( ˜X/X)
§kgv†vtvuˆYt‰…Bu³k…Šnv†iˆY‚jƒ‚zpYgpiƒnptf‚€™fg£wBgsidnsjƒjƒgpt‰gYc
9A@¡ ¢bd¤£¦¥AisD Galois
¡
‰¶ˆYt€y F, K
ˆksVjƒgpt„gjd‚ K ⊂ F
§ktf—ptf‚¦t„… F
‚„‡†pgdu‡jƒ‡{ga‚zrŠxzwŠt‰gdˆYg‚t‰…Šq K
c
§ Ù ¨ …RA (xw¯ ¦fw 12 gz”}• F ‚ “ m ‡™”n„kgih ”‘l‘’”gˆv3rv„f”‘l8o E ‚ §
q ƒ¸l–’xr ‚ ”8h™€ Galois
lBr‡y p „
”8h™o ‚ “ m ‡™”n„kgih™oR„‰’‰”x˜™oj€B„ ‚Bƒ€n„k…8
Gal(F/K) = {σ ∈ Aut(F) : σ(k) = k ∀ k ∈ K}.
hX™v‡ˆYgdjd‚ttfgv†%…Šjƒes—fg Galois
†vggsrƒ…std‚€id‚‰‡t‰gƒuBgprŠ—¤‚zwB‚‰‡†v…Šqdytt„…ƒqdygsqpt„…ƒjƒ…Š™f‹TuˆYjƒ…Šnsyt„…ƒqEˆksVjƒgpt‰…Šy F
§Vrƒ…Šq‘gd‹‡Žp†p…ƒqp†‘ˆYgsjƒ‚‰u{gswBe‘ˆkt‰gp¡T‚€™f— t‰… ˆksVjƒg E
§Vjd‚ Gal(F/K) ≤
Aut(F)
c²u{g¥tfgv†¥ˆ‡qv†px€³k‚‰uŒga¡Tg¤rƒgd™d…Šqdˆ‡u{edˆY…Šqsjƒ‚ˆYnv†vt‰…ŠjƒgwBeprŠ…Bu{ggprŠ…ptf‚zidx‰ˆYjƒgpt‰gatfgsy¡G‚zyV™v‡{gsy
Galois
jƒ‚t”ƒesˆ‡gat‰…Šqsy5´•7PwBgdu‰5 x7 §‡jƒ‚hˆkwB…ƒrƒ—a†pg¥‹‡tfesˆY…Šqdjd‚Xˆkt‰…¤¡T‚‰jd‚ziŠusV—f‚€yX¡T‚€sV™dgdjdgtfgsym‚ž†Eid—pp‡… ¡T‚zyV™v‡{gdypc ¡
‰V†vg gp†vespkyRpY…”rŠ…piƒqps††pqsjƒ… f
…v†p…Šjƒesf‚zt‰gƒuV—vu{gs³kyV™f‡ˆTu{jd… gv†pTu{gDwBev¡G‚‘™v‡{fgDt‰…Šq¬™§tt‰… f (ρ) = 0
§X—fgpidgd—fŽ”gv† —f‚ž†Dx€³k‚‰u$™f‡Œd‚‰y‘rŠ…pisiƒgprƒid—stdgst‰gsyjƒ‚zpYgsidnstd‚€™fgdy¦tfgsyjƒ…v†pes—fgsysc
WEF‡a`vu ¨ (xw¯P¯ w 12 gi”¸• f(x) ∈ F[x] ‚Bƒ€B„†… m €B„ p …„t f• q ƒg…{rxld“lV‹’v0€†’xr‰lG‡‰„†… F ‚‘ƒ€n„k…
r ƒ„ ‚ “ m ‡™”n„kgih ”Bl–’”gˆv3rv„f”Blfo K ‚ y€ F ‚‘ƒ€n„k…v”Bl splitting field
”Bl–’ f
”–ex” ‚98
|Gal(F/K)| = [F : K].
§ Ù ¨ …RA (xw¯†° w 12 gz”¸• F
gˆv3ri„ ‚ yX€ G ⊂ Aut(F)
”xe–” ‚ 5
m ”Bl–’xr ‚ ”Bl
FG
= {f ∈ F : σ(f) = f ∀σ ∈ G}.
€B„ ‚Bƒ€B„†…”Bl gi”n„5
‚ q e gˆv3rv„ ”Bl–’ G
gz”Blf€ F ‚

d€ · !¡9£¦¥¨§¦¥©!#¥ ¡¦¥©! ¡¦¥© ¥! ¡£ Galois
u¦gd™fgptfgs™f…Šnsjƒ‚$—ptvuBt‰… FG ‚‰‡†pgduBxz†vg%qprŠ—ŠˆkyVjƒgit‰…Šq F
cVh rƒgd™dgsrƒep†vyD…ƒ™vuˆYjƒ—Šy¦‚‰‡†vgƒuˆ‡gsjƒgv†vtfu{wŠ—Šy%—pt‰gp†¤t‰… G
‚‰‡†pgƒu6qprŠ…Š…Šjded—dg t€y†† Aut(F)
cu¦gd™dgdjdxz†v‚„u6—ŠjsyVygd“vu{—piƒ…ppY…ŠygswŠ—Šjƒg£wBgƒuYgp†‚‚‰‡†pgdu‡xž†pgagprƒid—¥qsrƒ…Šˆ‡nv†p…piƒ…#§d”diƒxzrŠ…v†vt‰gsyt—ptvuTgv†
H ⊂ G
tf—ptf‚ FG
⊂ FH
.
‰VrB‡ˆYgdy%gp† F
‚zrŠxzwŠt‰gsˆ‡g¤t„…ƒq K
wBgdu G = Gal(F/K)
§td—std‚ K ⊂ F G
⊂ F
c ehep¡T‚ˆYsVjdg‚gv†pesjƒ‚€ˆYghˆYt‰… K
wBgduƒt‰… F
…v†p…Šjdedf‚zt‰gduƒ‚ž†p—vu{edjd‚€ˆ‡…‚ˆYsVjdgYc¶l@tdgp†XrŠ‚€™v‡rƒt€yVˆYgrŠ…Šqg G
‚‰‡†pgduqsrƒ…Š…Šjded—fg¤tfgsy…Šjƒes—fgsyagsqst‰…Šjd…Š™f‹Guˆ‡jss††~t‰…Šq F
¡Gg©x€³k…Šqsjƒ‚£t„gmrƒgd™dgswBept€ygsrƒ…std‚ziƒx€ˆYjƒgpt„g
i
}† G
‚‰‡†pgƒuYjŠ‡{gqprŠ…Š…Šjded—dg£tdgdy Aut(F)
td—std‚ [F : FG
] = |G|.

ii
}† G, H
‚„‡†pgdu¶rƒ‚zrŠ‚€™fgsˆ‡jdxz†v‚‰y qsrƒ…Š…Šjded—f‚€y¥tfgsy Aut(F)
jƒ‚ FG
= FH §'td—std‚
G = H
c
WEF‡%ù ¨ (`w¯ ¥3w 12 gz”¸• F
€B„ ‚Bƒ€n„k…ˆr ƒ„ “ ‚ “ ‚ q „‰g‹r m €•h ‚ “ m ‡™”n„kgih”‘l‘’ gˆv3rv„f”‘l8o K
r ‚lBr‡y p„ Galois G = Gal(F/K)‚ „ m r ‚ e–”–… ‚Bƒ€B„†…r ƒ„ Galois ‚ “ m ‡™”n„‰gzh „†€X…|g8t ©
‚ …
r ƒ„„•“ej”–…|od“„ q „k‡‰y”¸• …|g™l p ©•€B„–r ‚ oegv’k€@5k˜‹‡ ‚ o 8

i K = FG
.
ii 3
y5
‚ „•€fy8’ˆ• ’l “lV‹’v0€†’xr‰l p(x) ∈ K[x]
w ”Bl l“•l ƒl mt ‚ … q ƒ
‚ o6gz”Bl F
w ‚Bƒ€B„†…
p …„t f• q ƒg…{rxl ‡‰„k… mt ‚ …veV ‚ oe”‘…|o q ƒ
‚ o ”Bl–’gi”Bljg™v‡rv„ F ‚
iii ¡ l†gˆv3ri„ F
„•“l–” ‚ ‚Bƒ ”Bl splitting field
‡‰y“•lx…|l–’ p …{„ tf• q ƒg‹…{r‰l‘’ “•ls’•%€ ©–r‰l‘’
f(x) ∈ K[x]‚
(xw„ £¢¡ H WEF ¨ FY1†„aªGFYA WEF‡a`vu ¨ 5u†A WEF‡#a`Q†A Galois
WEF‡%ù ¨ (`w¯R» % WEF ¨ Fk1R„%ªGF‡A 'XF‡%ù ¨ 5u†A 'FY#a`QUA Galois
) w
12 gz”}• F
“ ‚ “ ‚ q „‰g‹r m €xl–’ v„5†rxl©
‚ “ m ‡™”n„‰gzh Galois ‚ “ ƒ ”Bl–’ K
w‰r ‚ lBr‡y p „ Galois
”8h‹€ G = Gal(F/K)
w K = F G ‡k„k… E ‚ € p …{yxr ‚ g™l‘g™v‡rv„w p h8k„ p ˜ K ⊂ E ⊂ F ‚
¡ e–” ‚ ’ˆ“y qt ‚ …ur ƒ„ ~f€V~ „•€•”‘…|gi”Bl8…t ƒ„ r ‚ ”n„ “ ©
8
es•%€ ”¸•0€ ‚ € p …{yxr ‚ gk•0€ gk•‡r‡y”¸•%€ ”8h™o
‚ “ m ‡™”n„kgih™oe‡‰„†…es•%€”}•%€”’ˆ“•lflBr‡y p •%€X”8h™o G
wu“l–’ p ƒ€ ‚ ”n„†…i„“•e”8h‹€ E → Gal(F/E)
m ”xg‹…v%gz” ‚98
i §
g8t ‚ ”–…s‡‰efo v„65•r3efo”8h™o ‚ “ m ‡™”n„‰gzh™o p ©kl ‚ € p …{yxr ‚ gk•0€Žgk•3r‡yf”}•%€ ‚Bƒ€B„†… ƒg™lfo r ‚”Bl8€Rg8t ‚ ”‘…s‡ê p ‚Bƒ‡™”xh”¸•0€P„•€•” ƒgi”Bl8…tf•0€d’ˆ“lflBr‡y p• 0€R”xh‹o Galois
lBr‡y p „ko ‚ £ ’’‡ ‚ €
‡q …{r m €n„ h Gal(F/K)
w mt ‚ …”‘y “ h [F : K]‚
ii ˆ F ‚Bƒ€B„†…3r ƒ„ Galois ‚ “ m ‡™”n„‰gzh ‚ “ ƒ ‡‰y65
‚‚ € p …{yxr ‚ g™l–’ gˆv3rv„f”‘l8o E
wP„‘†iy ”Bl
E ‚Bƒ€B„†… Galois ‚ “ m ‡™”n„‰gzh ‚ “ ƒ ”Bl–’ K
„†€€ h„†€” ƒgi”Bl8…tGh ’ˆ“•lflBr‡y p„ Galois8
Gal(F/E) ‚Bƒ€B„†…r ƒ„‡‰„•€xlf€8…s‡ˆ˜6’ˆ“lflBr‡y p „”8h™o G‚ £ 1 „‰’‰”8˜‹€d”8h™€“ ‚ q ƒ“•”¸•%gih 5•„
mtAl–’xr ‚98
Gal(F/K)/Gal(F/E) ∼= Gal(E/K).
• bžb£gv†vtvuˆYt‰…Bu³Y‡{grŠ™f…dwBnsrdtf‚‰uTgv†vtvuˆYt‰…Bu³Bs††vt‰gsyiˆ‡‚hwBev¡G‚i‚ž†p—vu{esjƒ‚€ˆY…mˆYsVjdg E
tfgv†
Galois
…Šjded—dg Gal(F/E) ≤ Gal(F/K)
cy€” ƒgi” q l Š „ 8
}†vtvuˆYt‰…Bu³Bs˜ˆY‚awBev¡G‚qprŠ…ƒ…Šjƒes—fg H
tfgsy Gal(F/K)
t‰… ˆkt„gv¡T‚‰™d—EtfgsyˆYsVjdgˆYtdgp† F
§Y—fgpiƒgs—fŽ H → FH c

v ¡£ Galois ·
v¸
$† L, M
‚ž†p—vu{edjd‚€ˆ‡g ˆksVjƒgpt‰g tfgsy”‚zrŠxzwŠt‰gsˆ‡gsy K ⊂ F
wŠgƒu J, H
qsrƒ…Š…Šjded—f‚€y’tfgsy
Gal(F/K)
jƒ‚ H ≤ J
§Š¡Tgx€³By
F −→ 1 F ←− 1
∪ ∧ ∪ ∧
M −→ Gal(F/M) F H
←− H
∪ ∧ ∪ ∧
L −→ Gal(F/L) F J
←− J
∪ ∧ ∪ ∧
K −→ Gal(F/K) K ←− Gal(F/K)
$ ¨'¨ (`w¯ 3w 12 gi”¸• F ‚ “ m ‡™”n„kgihjg™v‡rv„”Blfo K
wzr ‚T‚ € p …y–r ‚ g†„dgˆv3rv„f”B„ L, M ‚ 12 gz”¸•
H, J
’ˆ“•lflBr‡y p ‚ op”8h™o Gal(F/K) = G‚ ¡ e–” ‚98

i Gal(F/F) = 1
‡‰„†… Gal(F/K) = G
w
ii F1
= F
w

iii yX€ L ⊂ M =⇒ Gal(F/M) Gal(F/L)
w
iv H J =⇒ FJ
⊂ FH w
v L ⊂ FGal(F/L) ‡‰„k… H Gal(F/FH
)
w
vi Gal(F/L) = Gal(F/F Gal(F/L)
)
‡‰„†… FH
= FGal(F/F H
)
‚
‰¶isrB‡{dy —stfuYt‰…rŠgs™fgpwBest€yDpY™fes‹Tgsjƒg‚¡Tggsrƒ…ŠˆYgd‹‡gp†s‡ˆ‡‚‰uŠtfqs³k—p†%gsrƒ…Š™v‡{‚€y
F −→ 1 −→ F −→ 1
∪ ∧ ∪ ∧
M −→ Gal(F/M) −→ F Gal(F/M)
−→ Gal(F/M)
∪ ∧ ∪ ∧
L −→ Gal(F/L) −→ F Gal(F/L)
−→ Gal(F/L)
∪ ∧ ∪ ∧
K −→ Gal(F/K) −→ F G
−→ G
l'³kŽsjƒg~bsc±R’‡{ga‚zrŠxzwŠt‰gsˆ‡g F
‚zrB‡kt‰…Šq K
rƒ…Šq%—f‚ž†%‚‰‡†vgƒu Galois
c
• rƒgd™dgsrƒep†vy ‚zrŠxzwŠt‰gsˆ‡g F
‚‰‡†pgƒu Galois
‚zrB‡6t„…ƒq K
gp†d† FG
= K
c ¡
hXjd…Bu{gmg F
‚„‡†pgdu‡‚zrƒx€wƒt„gsˆYg Galois
‚zrŠ‡Ytfqs³kgƒ‡…Šq‚‚ž†p—fuŒesjƒ‚€ˆY…ŠqaˆksVjƒgpt‰…Šy E
§Ygp†f† E = F Gal(F/E) c
§ `U ¨ …RA (`w¯ ˆfw 12 gz”¸• X ‚ € p …{yxr ‚ g™lTgˆv3rv„e”8h™o ‚ “ m ‡™”n„‰gzh™o K ⊂ F
A ˜ r ƒ„e’ˆ“lflBr‡y p „
”8h™o Galois
lBr‡y p „koj”8h™o ‚ “ m ‡™”n„‰gzh™oBwx„•€•” ƒgi”Bl8…t%„ ‚C‚ §
q ƒ¸l–’xr ‚ ”Bl X
€B„ ‚Bƒ€B„†…‡x ‚ …|gi”–e
„†€€ X = FGal(F/X) A
X = Gal(F/FX
Cf‚Cf‚ „q „”8h q l©–r ‚ e–”–…`h F ‚‘ƒ€n„k…ir ƒ„ Galois
‚ “ m ‡™”n„‰gzh ‚ “ ƒ ”Bl–’ K
w`„•€€X”Bljgˆv3ri„ K ‚Bƒ€B„†…v‡x ‚ …|gi”–e ‚
$ ¨'¨ (`w ° 3w 12 gz”¸• F
€B„ ‚‘ƒ€n„k… ‚ “ m ‡™”n„kgih ‚ €fefo6gˆv3rv„f”‘l8o K ‚EŒ “fy qt ‚ …fr ƒ„†~8€~
„†€”–…|gz”Bl8…t ƒ„ „†€8yxr ‚ gk„6”}•%€ ‡x ‚ …|gi”nv0€ ‚ € p …{y–r ‚ gk•0€Žgk•3r‡yf”¸•0€ ”xh™o ‚ “ m ‡™”n„kgih™oj‡‰„k…x”¸•0€
‡x ‚ …|gz”Bv%€Ž’ˆ“•l8l‘r‰y p •%€”8h™oelBr‡y p„ ‰o Galois
wf“•l‘’ p ƒ€ ‚ ”n„†…z„“•e E → Aut(F/E)‚
l'‚}jŠ‡{g Galois
‚€rƒxzwŠt‰gdˆYg—piƒgt‰gi‚ž†p—vu{edjd‚€ˆ‡giˆksVjƒgpt„gi‚‰‡†pgdudwŠid‚„uˆYtdewBgƒuŠˆktfgv†rƒ‚€™v‡
rƒt€yVˆ‡grƒ…Šqhgh‚zrŠxzwŠt‰gsˆ‡gt‚„‡†pgdudrƒ‚€rƒ‚€™fgdˆYjƒxž†p…ƒq¦”ƒgp¡Tjƒ…Šn©§s—sid‚€y …BuƒqprŠ…ƒ…Šjƒes—f‚€y'tfgsy}…Šjded—fgsy
Galois
‚‰‡†pgƒuY‚zrB‡ˆYgdy¦wŠid‚„uˆYtdx‰ypcRu¦™fepp‡jdgstfu

Uniformization for algebraic curves (in Greek)

Uniformization for algebraic curves (in Greek)

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Uniformization for algebraic curves (in Greek)