The document discusses Galois Theory and its historical context, particularly focusing on Évariste Galois, who made significant contributions to group theory and demonstrated why certain equations cannot be solved using radicals. It includes examples of group properties, the classification of symmetries, and the evolution of equation theory from ancient times to Galois' work in the 19th century. Additionally, it touches on Galois' tumultuous life and political involvement, leading to his early death at the age of 20.

1. Galois theory
A not-very-serious taster
x =
−b ± b2
− 4ac
2a
ax2
+ bx + c = 0
x = 3
−
q
2
+
q2
4
+
p3
27
+ 3
−
q
2
−
q2
4
+
p3
27
x3
+ px + q = 0
x5
+ ax4
+ bx3
+ cx2
+ dx + e = 0

2. • Invented (a substantial chunk of) group theory
• In particular, showed why some equations can’t be solved using roots
• Political
fi
rebrand, supported July Revolution in France (1830)
• Killed in a duel on May 31, 1832 (age 20)
Évariste Galois
Age ~15
Posthumous portrait by brother
Alfred Galois, age ~20
"Après cela, il y aura, j'espère, des
gens qui trouveront leur profit à
déchiffrer tout ce gâchis."
"Ne pleure pas, Alfred! J'ai besoin de tout
mon courage pour mourir à vingt ans!"

3. • A set of operations with the following properties…
• Closed
• Product of any two operations is still in the group
• Associative
• Ordering of brackets is unimportant: f(gh) = (fg)h
• NOT the same as saying that fg=gf (that’s if the group is commutative)
• Identity
• There’s an element that, when applied to any other element, leaves it unchanged
• Inverse
• For any element, there’s an element that “undoes” it (yielding the identity)
• Related concepts
• Ring,
fi
eld, etc: a set with two operations, akin to addition and multiplication
• Semigroup, monoid, etc: like a group, but with no inverse (e.g. string concatenation)
What is a group?

4. Symmetry (permutation) groups
|S4 | = 4! = 24
1 2 3
2 3 1
3 1 2
1 3 2
2 1 3
3 2 1
|S3 | = 3! = 6
1 2 3 4
2 3 4 1
3 4 1 2
4 1 2 3
2 1 4 3
4 3 2 1
1 4 3 2
3 2 1 4
2 1 3 4
3 2 4 1
4 3 1 2
1 4 2 3
1 2 4 3
3 4 2 1
4 1 3 2
2 3 1 4
4 2 3 1
1 3 4 2
2 4 1 3
3 1 2 4
3 1 4 2
1 3 2 4
2 4 3 1
4 2 1 3

5. Dihedral groups
1 2 3
2 3 1
3 1 2
1 3 2
2 1 3
3 2 1
D3 ≅ S3
} Rotations
Identity
} Re
fl
ections
The rotations (with the identity) form a cyclic subgroup of order 3
1 2 3
2 3 1
3 1 2
ℤ3
Rotations and re
fl
ections of regular polygons

6. Dihedral groups
D4 < S4
1 2 3 4 Identity
2 3 4 1 Rotation (90° CCW)
3 4 1 2 Rotation (180°)
4 1 2 3 Rotation (90° CW)
2 1 4 3 Re
fl
ection (north-south axis)
4 3 2 1 Re
fl
ection (east-west axis)
1 4 3 2 Re
fl
ection (NW-SE axis)
3 2 1 4 Re
fl
ection (NE-SW axis)
2 1 3 4
3 2 4 1
4 3 1 2
1 4 2 3
1 2 4 3
3 4 2 1
4 1 3 2
2 3 1 4
4 2 3 1
1 3 4 2
2 4 1 3
3 1 2 4
3 1 4 2
1 3 2 4
2 4 3 1
4 2 1 3
S4 = all permutations
D4 = rotations & re
fl
ections
The order of S4 is 24
The order of D4 is 8,
which is a factor of 24
All subgroups of a group G have an
order that is a factor of G’s order
D4 splits S4 into 3 cosets
Rotations and re
fl
ections of regular polygons
D4
S4

7. Even vs odd permutations
A3 ≅ ℤ3 < S3
S3
The "alternating group" contains
only even permutations
1 2 3 Identity Even 1 2 3
2 3 1 (12)(23) Even 2 3 1
3 1 2 (13)(23) Even 3 1 2
1 3 2 (23) Odd
2 1 3 (12) Odd
3 2 1 (13) Odd
Decomposition using
transpositions

8. Even vs odd permutations
1 2 3 4 Identity Even 1 2 3 4
2 3 4 1 (12)(23)(34) Odd
3 4 1 2 (13)(24) Even 3 4 1 2
4 1 2 3 (14)(24)(34) Odd
2 1 4 3 (12)(34) Even 2 1 4 3
4 3 2 1 (14)(23) Even 4 3 2 1
1 4 3 2 (24) Odd
3 2 1 4 (13) Odd
2 1 3 4 (12) Odd
3 2 4 1 (13)(34) Even 3 2 4 1
4 3 1 2 (14)(23)(34) Odd
1 4 2 3 (24)(34) Even 1 4 2 3
1 2 4 3 (34) Odd
3 4 2 1 (13)(24)(34) Odd
4 1 3 2 (14)(24) Even 4 1 3 2
2 3 1 4 (12)(23) Even 2 3 1 4
4 2 3 1 (14) Odd
1 3 4 2 (23)(34) Even 1 3 4 2
2 4 1 3 (12)(24)(34) Odd
3 1 2 4 (13)(23) Even 3 1 2 4
3 1 4 2 (13)(23)(34) Odd
1 3 2 4 (23) Odd
2 4 3 1 (12)(24) Even 2 4 3 1
4 2 1 3 (14)(34) Even 4 2 1 3
A4 < S4
S4
The "alternating group" contains only even permutations

9. • Group properties
• Closure: fg ∈ G
• Associativity: f(gh)=(fg)h
• Identity: g1 = 1g = g
• Inverse: g-1 ∈ G
• Numerical groups
• Integers, under addition
• Integers modulo N, under addition
• Nonzero rational numbers, under multiplication
• Nonzero complex numbers, under multiplication
Number groups
(ℤN, + )
(ℤ, + )
(ℚ∖{0}, × )
(ℂ∖{0}, × )
Cyclic group

10. Quaternions
i2 = j2 = k2 = ijk = -1
Currently the standard way of doing 3D
rotations in computer games & graphics
.
Also v important in physics

11. Rubik’s Cube
The Rubik's Cube group, G, is de
fi
ned to be
the subgroup of S48 generated by the 6 face
rotations, { F, B, U, D, L, R }.

12. Cryptanalysis of Enigma
Marian Rejewski (d.1980)

13. Every symmetry
of a physical
system has a
corresponding
conservation
law.
Time invariance
Conservation of
energy
Translational
invariance
Conservation of
momentum
Rotational
invariance
Conservation of
angular momentum
Noether’s Theorem
(1918)

14. • Recall that if H is a subgroup of G, then H's order is a factor of G's order,
and H splits G into equally-sized cosets
• This is analogous in a certain way to "dividing" G by H.
We can de
fi
ne a quotient group, G/H, which represents H's cosets
• In some sense this is like factoring out a component of G's symmetry
• A group is called solvable if it can be broken down completely in this way
(some technical details: the subgroups must all be normal subgroups - which means
the left & right cosets are equal - and the quotient groups all have to be commutative)
“Solvability” of a group
ℤ3 ≅ A3
S3 ≅ D3 D4
S4 A4 ℤ4
S3/A3 ≅ ℤ2
K
Klein 4-group
S4/A4 ≅ ℤ2
A4/K ≅ ℤ3

15. • S3 (3!=6 permutations) is solvable
• S4 (4!=24 permutations) is solvable
• S5 (5!=120 permutations) is not solvable
“Solvability” of a group
1 < ℤ2 < S3
1 < K < A4 < S4
The “solvable” terminology relates to the problem of solving equations….

16. Quadratic Equations
Equation Theory
https://www.mathnasium.com/the-history-behind-the-quadratic-formula
~1500 BCE
Ancient Egypt
~400 BCE
Babylon, China
~300 BCE
Ancient Greece
x =
−b ± b2
− 4ac
2a
ax2
+ bx + c = 0 ⇒
(
x +
b
2a)
2
=
b2
4a2
−
c
a
Pythagoras Euclid
Muhammad ibn
Musa Al-Khwarizmi
~700-800 CE
India, Persia
Brahmagupta
~1500 CE
Girolamo Cardano
Lookup tables
Completing
the square
Irrational
numbers Zero
Imaginary
numbers
Completing
the square
2019 CE
Po-Shen Loh
(yeah right)

17. • Bologna University public mathematics
competitions
• Tartaglia vs Scipione del Ferro (1535)
• Publication in verse by Tartaglia (1539)
• Publication by Cardano (1545)
He also published the solution to the quartic
Cubic Equations
Equation Theory
Cardano’s Oath: I swear to you, by God’s holy Gospels, and as a true man of honour, not only never to
publish your discoveries, if you teach me them, but I also promise you, and I pledge my faith as a true
Christian, to note them down in code, so that after my death no one will be able to understand them.
x3
+ ax2
+ bx + c = 0

18. • Despite rapid progress from cubic to quartic, and great interest among the
mathematical community, no-one solved the quintic in the next 150 years
• Circa 1770, Lagrange started looking at groups of permutations and
functions that were invariant under those permutations
• In 1799 Paolo Ru
ffi
ni proved it was impossible to
fi
nd a general formula to
the quintic as shown above, using “radicals” (arithmetic + n’th roots)
• Ru
ffi
ni’s proof had a hole, which was plugged by Niels Henrik Abel in 1824
• Galois went further, and showed which speci
fi
c equations can be solved.
For example, this one can be solved:
• I can give some
fl
avor (not rigorous), but
fi
rst let’s look at who Galois was
Quintic equations
x5
+ ax4
+ bx3
+ cx2
+ dx + e = 0
x5
− 1 = 0
Footnote: equation theory might all seem a bit quaint nowadays, when we’d just use numerical methods to
fi
nd the roots to arbitrary
precision in a matter of nanoseconds. Still, mathematicians like to understand things deeply, and that leads to some nice things.

19. A short history of the French Revolution
In 1972, Chinese premier Zhou Enlai was asked about the impact of
the French Revolution. “Too early to say,” he replied.
Estates General of 1789
initially convened by Louis XVI
in response to parliamentary unrest
Evolves into a people’s assembly
Proceeds to abolish feudal system
Mobs storm palace, Bastille
1789
1792
Execution of Louis XVI
for conspiring with forces
outside France.
Start of “Reign of Terror”
Execution of Robespierre
End of “Reign of Terror”
1794
1799
Napoléon Bonaparte takes power in a
coup. Starts spreading Revolutionary
principles across Europe by force
1812
Napoléon retreats from Russia in defeat
Eventually exiled to Elba by the British
1814
Bourbon Restoration
Louis XVI’s younger brothers
each take turns at trying to
undo the Revolution
1830
July Revolution
1811
Galois born
1832
Galois dies
…tensions…

20. • Parents were well-educated. Father supported the Revolution, became town mayor in 1814 (after
Bourbon restoration). Opposed by conservative forces including local priest
• Évariste taught himself mathematics from Legendre’s Éléments de Géométrie, quickly left teachers
behind, started publishing original research papers. Untidy work, impatient, genius di
ffi
cult student
• In 1829 his father committed suicide after a scandal maliciously engineered by the priest. Évariste
deeply a
ff
ected. “I've lost my father and no one has ever replaced him, do you hear?”
• Expelled after eviscerating school Director in a letter to Gazette des Écoles following July Revolution
(1830). “The same day, M. Guigniault told us with his usual pedantry: ‘There are many brave men
fi
ghting on both sides…’ There is the man who the next day covered his hat with an enormous tricolor
cockade.”
• Joined Artillery of the National Guard (anti-monarchist militia), arrested after proposing a toast to the
new King’s execution. Acquitted at trial. “To Louis-Philippe, if he turns traitor!”
• Arrested again after leading a protest, armed to the teeth. Sentenced to 6 months. In jail, got drunk for
the
fi
rst time; confessed his depression, predicted own death in duel, tried to kill himself
• Love a
ff
air with someone (daughter of physician?). Did not work out. “How can I console myself when
in one month I have exhausted the greatest source of happiness a man can have?”
• Shot at point-blank range in what may have been more like Russian Roulette than a duel. Opponent
unknown for sure. Abandoned by seconds, found by a farmer. Died of peritonitis the next day.
• Refused last rites. Last words (to brother): “Don't cry, Alfred! I need all my courage to die at twenty!”
A short biography of Évariste Galois

21. (correct technical term is “conjugacy”)
“Indistinguishability”
Note that (+i)2 = -1
…but also (-i)2 = -1
Choice of +i and -i is arbitrary
Can't distinguish +i and -i
…there is a symmetry between them
Argand diagram of complex plane
Real
Imaginary
+1
-1
+i
-i

22. • The Galois group of a polynomial is the set of “indistinguishable”
permutations of the roots
• “Indistinguishable” means that any(*) equation satis
fi
ed by the
roots will still be satis
fi
ed when the roots are permuted
(*) the equation must have rational coe
ffi
cients
Galois group of a polynomial
x5
= 1 1, ω, ω2
, ω3
, ω4
f(1, ω, ω2
, ω3
, ω4
) = 0
f(a, b, c, d, e) = b2
− a
For example, the equation has roots
Now, if we de
fi
ne then
but also f(1, ω4
, ω3
, ω2
, ω) = 0
f(1, ω2
, ω4
, ω, ω3
) = 0
but note f(ω, ω2
, ω3
, ω4
,1) ≠ 0
f(1, ω2
, ω, ω3
, ω4
) ≠ 0
Calculating the Galois group of a polynomial is hard!
Note that it’s de
fi
ned in terms of the roots, not the coe
ffi
cients.
This sunk Galois’ chances of winning the Grand Prize in Mathematics.
Poisson declared his manuscript “incomprehensible”…
ω = e2πi/5

23. • Equations that are solvable by radicals (i.e. +, -, *, /, and √ )
have solvable Galois groups
Fundamental Theorem of Galois Theory
x5
+ ax4
+ bx3
+ cx2
+ dx + e = 0
(x − r1)(x − r2)(x − r3)(x − r4)(x − r5) = 0
r1r2r3r4r5 = − e
r1r2r3r4 + r1r2r3r5 + r1r2r4r5 + r1r3r4r5 + r2r3r4r5 = d
r1r2r3 + r1r2r4 + r1r2r5 + r1r3r4 + r1r3r5 + r1r4r5 + r2r3r4 + r2r3r5 + r2r4r5 + r3r4r5 = − c
r1r2 + r1r3 + r1r4 + r1r5 + r2r3 + r2r4 + r2r5 + r3r4 + r3r5 + r4r5 = b
r1 + r2 + r3 + r4 + r5 = − a
Suppose this general quintic equation
can be factorized using its
fi
ve roots
Then the roots satisfy certain symmetric polynomial equations:
The Galois group of this polynomial is S5. This group can't be factorized (i.e. is not solvable). Factorizing the equation is like
factorizing the group, in a sense that can be made explicit with reference to the
fi
eld that contains the roots, and how this
fi
eld
is constructed using radicals. Since S5 is not a solvable group, there is no equation to solve the general quintic (using
radicals). This result was proved by Ru
ffi
ni (1799) and Abel (1824), but today is generally described using Galois theory.

24. • Group theory is a foundation of pure math & is all over modern science
• Physics: fundamental laws, subatomic particles, dynamical systems, quantum states…
• Chemistry: chirality, conformation enumeration, conjugacy classes…
• Biology: crystallography, viral capsids, systems of ODEs in math.bio…
• Computer science: coding theory, cryptography, category theory/monads, graphs, complexity…
• What are we to make of Galois himself?
• Tendency to romanticize his story, exaggerate the “misunderstood genius” angle (or to speculate
that police spies conspired to kill him, etc.)
• The idea that he was misunderstood by mathematicians of the time has some truth, but there were
several who understood the profundity of his work
• Historians seem to agree he was deeply a
ff
ected by his father’s death. This went on for years.
Could be “complicated grief” / prolonged grief disorder. The more you read, the sadder it is
• Given his own 100% correct prediction that he’d die in an “a
ff
aire d’honneur”, and other facts, his
fate reads a lot like suicide-by-duel. Romanticizing this is inadvisable. Still a fascinating story.
• What about the impact of the French Revolution?
• I agree with Zhou Enlai: it’s too early to say
Closure