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Les polynômes formels à une indéterminée à coefficients dans un corps K

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Groupes, Permutations, Anneaux, Arithmétique dans Z, Corps commutatif, Les polynômes formels à une indéterminée à coefficients dans un corps K, Fonctions polynomiales, racines, Espaces vectoriels, K-algèbres, Espaces vectoriels de type fini, Matrices, Déterminants, Fractions rationnelles, Produit scalaire sur un R-ev, Espace vectoriel euclidien, R-ev euclidien orienté de dimension 2, R-ev euclidien orienté de dimension 3, Espaces affines, Géométrie dans un espace affine euclidien

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Les polynômes formels à une indéterminée à coefficients dans un corps K

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  2. 2. 6 ! ) × × 1 1 % × 1 $ 7 , ×# + '''$.#.#%= ( ∈= $% 1 . = .# =≥∀ ' * ∈= ∈$% ' ∈= $% # 2 =×=== =+ . " = # 1 = # ×' " 1 × 0 * ∈=== ∈∈∈ $%#$%#$% ' ∈=+× $%$% # 2# ∈ =+=+ + ↑ =+ ↑ =+ +=+=+= 1 #11 $% ( ∈=×+× $% # 2# ∈ =+=+ += ' " ×+×=+× $% ( ×+×=×+×=+×=×+ $%$% 1 × * ∈=== ∈∈∈ $%#$%#$% ' ∈=× $% # 2# ∈ # =+ = ' ( ∈=×× $%$% # 2# ∈ # =+ = " =++ + ↑ =+ =+ ↑ =+ =+ === 1 11 # 8 ∈=×× $%$% # 2# ∈ =++ = ' "+2 ××=×× $%$% # × 1 ' " $##% ×+ ' 9$ ( 6 ' * λλλφλ φ :#'''$.#.#%$% = → ' φ 3 + , $%$%'''$.#.#%'''$.#.#%'''$.#.#%$% µφλφµλµλµλφ +=+=+=+• $%$%'''$.#.#'''$%.#.#%'''$.#.#%$% µφλφµλλµλµφ ===• touscours.net
  3. 3. ; ! < , ∈= ∈∈ $%'''$.#.#%'''$.#.#% $%$% µλ 1 =+ = ' " λµ== .. .= .= ' #'''$.#.#%$% ==• φ • ( # '''$.#.#%'''$.#.#% µλ = # 1 µλ = & # $%φ # + $##% ×+ ' %/ & += >$ / 1 # - ) ) # & ∈λ # λ λ: ' # ∈µλ# ∈# % 1 ∈= $%:λ # ∈= $% $ ∈===• $%#'''$.#.#%: λλλ 2 λ== =+ " ∈== $%#'''$##% 6. λλλλλ µλµλ +=+• $% λλλ +=+• $% $%$% µλλµ =• =×=×=ו : ' $ ( ; + * + #'''$.#.##.%= - ) ) ∈= $%δ 2 =δ {} .#? =∈∀ δ ∈=∈∀ $%# $% 2 $% = { } .#? $% =∈∀ " ' .= '''$.#.#%. == * ∈ # & ∈= $% $% 2 $% = { } .#? $% =∈∀ ∈ + == $% /2 == ≠ ==∈∀ − =+ .. . # . $% . $% $% δ δ δ βα βα * # .≠ # += +≠ == − .$% " ∈ ++ = $% $% @ * ∈ ' + & ∈ = 2 # + ' touscours.net
  4. 4. A ! " * ∈ ' • + ∈= $% # + + # + ' / +++= +++= +++= = ''' '''$.#.##.#'''.#.%''''''$.#.##.%'''$.#.#% '''$.#.##.#'''.#.%''''''$.#.##.%'''$.#.#% '''$.#.##'''#% . . . . . "+2 +, ∈ 2 + ' • B + * ∈∈ = # 2 + ' ∈ ∈∈ ∈ === $%$% ' C ) 3 ∈ # 2 + % ∈$% $' 7 ' ) + ' ) + DE ' "$ ( A # • @ DE∈ + & ∈ = 2 + ' # =∈∀= ∈∈ # • $#D#E% ×+ # ' • * ∈ = # ∈ = 2 + + # = = . # = = . ==== +=+=+=+ .... $% # 2 $#,%= touscours.net
  5. 5. F ! [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] + = + = =+ ∈∈ + = =+ ∈∈ + ∈ ∈ + ∈ ∈== === ===× .. #.##.. #.##. #. #. #. #... $$%% /2 =+ = $ " * DE∈ # ∈ = 2 + ' • * DE.= # - ) ) .= .= ' .# =∈∀ ' / & ∞− • * # DE.≠ , ∈ & .≠ ' / { }.#, ≠∈= % & + 1 # 3 + $' ' DE∈ # { }∞−∪∈ # +& 1 ∈⇔≠ . DE . , { }.?∈λ , + 1 .≠ , =. 1 .≠ ' ≤ , =. + DE G # DE. + & # ' 9$ * DE# ∈ ∈λ ' $$%H#%$%H# . $% $$%##%$% $#,%$% −∞=−∞×∈∀=∈∀• ∞− ≠ =• −∞=−∞+−∞−∞=+−∞∈∀+=ו ≤+• λ λ " ) .== 1 ' touscours.net
  6. 6. ! ) * .= .≠ % .≠ .= $# ' ) 4 .≠ .≠ 5 = = ' • / $#,%= ' / = = . # = = . " = +=+ . $% # ≤+ $% =ו == .. 2 .≠ .≠ " ''' . +=× + ≠ = =• . λλ 1 .≠ • I 1 ' C ) * # + ' ) 1 # .' ) + # + J ' ) B & ' $#D#E% ×+ @ $#D#E% ×+ ' " ) " 3# $#D#E% ×+ { }. ' ) * DE# ∈ # & .= 4 & + .. == ' * & # - ) ) & .. ≠≠ ' * # == ' # ∈# ' = = . = = . # 1 .≠ .≠ 4 + .≠ # .≠ # & , ' $% += # −∞=+ # −∞= # −∞= ' touscours.net
  7. 7. K ! @ 1 $#D#E% ×+ , { }.? ' " • * DE∈ ' * 1 # , DE∈ & = ' .=+ ' / # { }∞−∪∈# ' " .== • I & # { }.?∈= λ # λ 1 − λ ' " − = λ 1 ' " * DE# ∈ ' / & & 1 & + , DE∈ & = ' " * DE# ∈ ' / & & 1 1 % & & 1 & + , { }.?∈λ & λ= $ " • * λ= 1 { }.?∈λ 1 − = λ 1 ' • * 1 1 , DE-# ∈ & = -= ' " -= ' " .$-% =− # +2# DE # .= -= ' o * .= # .= 1 , DE∈ & .== ' o * -= # 1 # { }.?∈= λ # λ= ' $ " 1 DE ' @ * DE# ∈ # 1 .≠ ' , & $#% + DE & += $%$% < ' / & & 1 & 1 ' " • B * += # 1 $%$% < ( --+= # 1 $%$-% < # −=− -$-% ' " $-#,% $-%$-%$% ≤ −=−+ # $%$-%$% <−+ # ∉− $-% ' " .-=− ' " −=× -. # =- ' "+2 + ' touscours.net
  8. 8. L ! • (, * { }.?DE∈ # ∈ ' 4 & $%#∈∀ # 2 <+=∈∃∈∀= #DE$#%D#E$% 6 ) " 3# $.% 1 # & .≤ # o * ≥ # +×= . # $#.% 1 ' o * .= # { }.?∈= .$% += − # $.#% − 1 ' ) * ∈ # $% ' * +≤ ' + ≤ + ∈ + +++= 2 . .''' ' * +≤ % # +×= . $#.% 1 $ / $% +−+= + + < −+− + −+− + " ≤ −+− + −+− + −+= 66 # $% / # ≤6 ' " 6 += 1 6 DE$#% ∈ <$% ' " $% ++= −+− + ' (, 666 6;F +=−−+= 6;$%66$% 66666$% ;6; ;;;;6; −−+=−−−++= −−+=−−+−+= ! " $ " # * DE∈ ' + ∈ = 2 + # + ' " = = . * DE∈ ' / ∈ =$%: # = = . $%: @ * DE# 6 ∈ DE∈ # DE∈λ ' λλ = ×=× +=+ $%: $%:$%:$$%:% $%:$%:$$%:% 66 66 touscours.net
  9. 9. ! ! " * ∈ = # ∈ =6 ' / ∈ & ≤ ≤6 ' / = +=+• . 6 $% " $$%:%$%$%:$%: 6 ... 6 +=+=+=+ === = =ו 6 . 6 2 =+ = == =+≤≤ + == ====× 6 . 6 .#... 6 $%:$%: λλλλ ===• .. '$%':$%: I & + DE∈ # + $%: DEDE → + $#D#E% ×+ % 3 3 $' I & DE.∉ # $%: 3 1 ' 9$ # / & .≠+ % - ) ) & + + 6 $#% + $ " * DE∈ / & & $%$%$% ==− / & & $%$%$% −=−=− .# 6 =∈∀ + .# 6 =∈∀ " ∈ −=− $%$% ' .'6# #$%#" 6 66 =∈∀⇔ =−∈∀⇔=−∈∀⇔ + ++ / # 6666 $%'6 ++++ +=+= ' / & .≠+ ' " .#.'6# 66 =∈∀⇔=∈∀ ++ / 8 ' touscours.net

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