Galois Theory - Introduction

Supriya Pisolkar

IISER Pune

August 8, 2019

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What are polynomials?

X + 1

X 2 + 2

3X 3 + 2X 2 − 5

In general a polynomial can be expressed as

anXn + an−1X

n−1 + · · ·+ a1X + a0

where ai are some numbers.

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Degree of a polynomial

The highest integer power n appearing in

anXn + an−1X

n−1 · · ·+ a2X2 + a1X + a0

is called the degree of a polynomial.

Type Example Degree

Linear X + 1 1Quadratic X 2 + 2X + 1 2

Cubic X 3 + 2X 3Quartic 2X 4 + x3 + 2 4Quintic X 5 + 1 5

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Egyptians and Babylonians : 2000 BC

Given a specific area, they were unable to calculate lengths of the sides ofcertain shapes, and without these lengths they were unable to design floorplan for their kingdom. In the process, Babylonians could solve pairs ofsimultaneous equations of the form: X + Y = s, and XY = t, which areequivalent to the quadratic equation X 2 + t = sX .

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Solving polynomials

Solving a polynomial p(X ) means finding numbers which whensubstituted in place of X give zero, also called finding roots of apolynomial.

e.g. X 2 + 1 has no root in R but has a root i =√−1 ∈ C.

Question

Given a polynomial p(X ), how do you find it’s roots? How do you find it’sall roots?

Let’s have a look at some small degree polynomials.

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Solving quadratic polynomials : 6th centuryBrahmagupta, an Indian mathematician, was the first one to computesquares, square roots, cubes and cube roots of integers. He realised that aquadratic equation can have two solutions and one of which could be anegative solution.

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Formula for roots of quadratics - 9th centuryMany centuries later, algebra was invented by a Persian mathematicianMuhammad al-Khwarizmi.

Based on the work of Brahmagupta, he wrote a book Hisab al-jabrw’almuqa-bala. He was the first, who gave a method of solving anyquadratic equation by reducing it to one of the six forms of equationsgiven by him. The word algebra came from the title of this book. Considera general quadratic aX 2 + bX + c then the roots are given by

x =−b + (

√b2 − 4ac)

2a, x =

−b − (√b2 − 4ac)

2a

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Cubic polynomials - 11th to 16th century

Many mathematicians were working on solving cubic equations. Significantefforts were made by Omar Khayyam ( Persia) and Fibonacci ( Italy) butwithout any luck to actually giving formula for roots. Gerolamo Cardano,an Italian mathematician first published the formula for roots of cubicpolynomials

Figure: Gerolamo Cardano (1501-1576)

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Roots of a cubic polynomials

Consider a cubic equation

aX 3 + bX 2 + cX + d = 0

Formula for a root of this polynomial is given by

x =3

√( −b3

27a3− bc

6a2− d

2a) +√

( −b3

27a3+ bc

6a2− d

2a)2 + ( c3a −

b2

9a2)3 +

3

√( −b3

27a3− bc

6a2− d

2a)−√

( −b3

27a3+ bc

6a2− d

2a)2 + ( c3a −

b2

9a2)3 − b

3a

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Quartic polynomial: 16th Century

What happens to quartic polynomials? The analogous formula for roots ofquartic polynomials was given by Lodovico Ferrari in 1545.

The formula is much worse than that for cubic (I won’t even try to write itdown here!! )For next 300 years, people were trying to solve quintic and higher degreepolynomials.

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Fundamental theorem of Algebra : 18th century

By using complex numbers, Carl Friedrich Gauss, a Germanmathematician, regarded as one of the greatest mathematicians of all time,

proved a following remarkable result(1799) : ‘Every non-constantpolynomial f (X ) ∈ C[X ] has a root in complex numbers’.

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Mysterious Quintic polynomials: 19th century

In 1824, Neils Abel, Norwegian mathematician proved that it is impossibleto solve a general equation of the fifth (or higher) degree in terms ofradicals.

But, this result did not settle the matter completely as this did not providenecessary and sufficient conditions for saying precisely which quintic (andhigher) equations are unsolvable by radicals. For e.g. one can always writedown the solutions of the equation X 5 − 1.

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Evariste Galois

During the same time, a French mathematician Evariste Galois ( who wasonly 18 years old at that time) gave the necessary and sufficient conditionfor a quintic polynomial to be solvable by radicals.

Allegedly he was in love with a woman engaged to an artillery officer.Galois challenged the officer in order to win her affection. In the nightsleading up to the duel, he wrote letters to his friends. The most significantof these had the ideas which are foundation of Galois Theory whichbrought to light non-solvability of a general quintic. He died at the age of21.

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A mathematician whose collected works run to a mere sixty pages, onewho did not live to be 21 and yet truly revolutionised modern mathematicsthrough his pioneering discoveries - this uniquely defines Evariste Galois! -B. Sury ( ISI, Banglore)

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Galois Theory : Construction of Splitting fields

Galois had another approach to the problem of quintic by using fields andgroups. In fact he created the theory of groups, and he was the first one touse the word “group” in it’s technical sense. Galois theory works for anyfield. Consider an irreducible polynomial p(X ) ∈ K [X ], where K is a field.

Example: X 2 + 1 is irreducible in Q[X ].

One constructs a field K (α) which is the smallest field containing K and aroot α of p(X ).

Example: Q(i) contains a root i of X 2 + 1. We say that Q(i) is obtainedby adjoining i to Q.

When we ‘adjoin’ all roots of p(X ) to K , then that ‘bigger’ field is calledthe splitting field of p(X ).

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Galois theory

Galois showed that, for a polynomial p(X ), there is a way to associate;

p(X ) ∈ K [X ]→ splitting field L/K → Galois group G(L/K)

The Galois group G (L/K ) is closely related to the group of automorphismsof L. The association is such that to study roots of a polynomial it isenough to study the corresponding group and vice-versa.Galois proved that roots of a general quintic polynomial can not beexpressed in terms of radicals whenever the associated group is‘non-solvable’. He could explicit construct examples of degree 5polynomials that are not solvable by radicals. One such polynomial isX 5 − 4X + 2 ∈ Q[X ].

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Applications of Galois Theory

What is astonishing is that Galois theory solves with equal ease,

1 an algebraic problem like the solvability by radicals of a quintic.

2 classical geometric problems like (i) to construct a square whose areais that of the unit circle, (ii) to trisect a given angle and, (iii) toconstruct a cube whose volume is twice that of a given cube, all ofwhich are to be done with the aid of a straight-edge and a compass.

3 fundamental theorem of algebra is a easy consequence of Galoistheory.

4 Another important discovery by Galois was the determination of thefinite fields.

5 The Galois Correspondence is a powerful idea which can begeneralized to apply to such diverse topics as ring theory, algebraicnumber theory, algebraic geometry, differential equations andalgebraic topology. Because of this, Galois theory in its manymanifestations is a central topic in modern mathematics.

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References

Algebra - S. Lang - Third Edition.Galois Theory : T.I.F.R Mathemtical Pamphlets - 3.Basic Algebra- Vol. 1 : Nathan Jacobson.Abstract Algebra - Dummit and Foote.

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Thank You!

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