Galois theory

MA3D5 Galois theory Miles Reid Jan–Mar 2004 printed Nov 2004

Contents 1 The theory of equations 1.1 Primitive question . . . . . 1.2 Quadratic equations . . . . 1.3 The remainder theorem . . . 1.4 Relation between coefficients 1.5 Complex roots of 1 . . . . . 1.6 Cubic equations . . . . . . . 1.7 Quartic equations . . . . . . 1.8 The quintic is insoluble . . . 1.9 Prerequisites and books . . Exercises to Chapter 1 . . . . . .

. . . . . . . . . . . . . . . . . . and roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

2 Rings and fields 2.1 Definitions and elementary properties . . . 2.2 Factorisation in Z . . . . . . . . . . . . . . 2.3 Factorisation in k[x] . . . . . . . . . . . . 2.4 Factorisation in Z[x], Eisenstein’s criterion Exercises to Chapter 2 . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

. . . . . . . . . .

. . . . .

3 Basic properties of field extensions 3.1 Degree of extension . . . . . . . . . . . . . . . . 3.2 Applications to ruler-and-compass constructions 3.3 Normal extensions . . . . . . . . . . . . . . . . 3.4 Application to finite fields . . . . . . . . . . . . 1

. . . . . . . . . .

. . . . .

. . . .

. . . . . . . . . .

. . . . .

. . . .

. . . . . . . . . .

. . . . .

. . . .

. . . . . . . . . .

. . . . .

. . . .

. . . . . . . . . .

. . . . .

. . . .

. . . . . . . . . .

. . . . .

. . . .

. . . . . . . . . .

. . . . .

. . . .

. . . . . . . . . .

3 3 3 4 5 7 9 10 11 13 14

. . . . .

18 18 21 23 28 32

. . . .

35 35 40 46 51

3.5 Separable extensions . . . . . . . . . . . . . . . . . . . . . . . 53 Exercises to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . 56 4 Galois theory 4.1 Counting field homomorphisms . . 4.2 Fixed subfields, Galois extensions . 4.3 The Galois correspondences and the 4.4 Soluble groups . . . . . . . . . . . . 4.5 Solving equations by radicals . . . . Exercises to Chapter 4 . . . . . . . . . .

. . . . . . . . . . . . . . . . . . Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

5 Additional material 5.1 Substantial examples with complicated Gal(L/k) . . . 5.2 The primitive element theorem . . . . . . . . . . . . . 5.3 The regular element theorem . . . . . . . . . . . . . . . 5.4 Artin–Schreier extensions . . . . . . . . . . . . . . . . . 5.5 Algebraic closure . . . . . . . . . . . . . . . . . . . . . 5.6 Transcendence degree . . . . . . . . . . . . . . . . . . . 5.7 Rings of invariants and quotients in algebraic geometry 5.8 Thorough treatment of inseparability . . . . . . . . . . 5.9 AOB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 The irreducibility of the cyclotomic equation . . . . . . Exercises to Chapter 5 . . . . . . . . . . . . . . . . . . . . .

2

. . . . . .

. . . . . . . . . . .

. . . . . .

. . . . . . . . . . .

. . . . . .

. . . . . . . . . . .

. . . . . .

60 60 64 68 73 76 80

. . . . . . . . . . .

84 84 84 84 84 85 85 86 86 86 86 87

1

The theory of equations

Summary Polynomials and their roots. Elementary symmetric functions. Roots of unity. Cubic and quartic equations. Preliminary sketch of Galois theory. Prerequisites and books.

1.1

Primitive question

Given a polynomial f (x) = a0 xn + a1 xn−1 + · · · + an−1 x + an

(1.1)

how do you find its roots? (We usually assume that a0 = 1.) That is, how do you find some solution α with f (α) = 0. How do you find all solutions? We see presently that the second question isQ equivalent to splitting f , or factoring it as a product of linear factors f = a0 ni=1 (x − αi ).

1.2

Quadratic equations

Everyone knows that f (x) = ax2 + bx + c has two solutions √ −b ± b2 − 4ac . α, β = 2a

(1.2)

Set a = 1 for simplicity. You check that α + β = −b,

and αβ = c,

(1.3)

which gives the polynomial identity f (x) = x + bx + c ≡ (x − α)(x − β). The relations (1.3) imply that ∆(f ) = (α − β)2 = (α + β)2 − 4αβ = b2 − 4c.

(1.4)

This gives the following derivation of the quadratic formula (1.2): first, the argument of §1.4 below (see Corollary 1.4) proves directly the polynomial identity x2 + bx + c ≡ (x − α)(x − β), hence equations (1.3–1.4). Thus we √ have an equation for α + β and for δ = α − β = ∆, that yield (1.2). The expression ∆(f ) of (1.4) is called the discriminant of f . Clearly, it is a polynomial in the coefficients of f , and is zero if and only if f has a repeated root. Over R, f has two distinct real roots if and only if ∆ > 0, and two conjugate complex roots if and only if ∆ < 0. Compare Ex. 15 3

1.3

The remainder theorem

Theorem 1.1 (Remainder Theorem) Suppose that f (x) is a polynomial of degree n and α a quantity.1 Then there exists an expression f (x) = (x − α)g(x) + c, where g(x) is a polynomial of degree n − 1 and c is a constant. Moreover, c = f (α). In particular, α is a root of f if and only if x − α divides f (x). Proof The “moreover” clause follows trivially from the first part on substituting x = α. For the first part, we use induction on n. Suppose that f (x) is given by (1.1). Subtracting a0 xn−1 (x − α) from f (x) kills the leading term a0 xn of f (x), so that f1 (x) := f (x) − a0 xn−1 (x − α) has degree ≤ n − 1. By induction, f1 (x) is of the form f1 (x) = (x − α)g1 (x) + const., and the result for f follows at once.  Corollary 1.2 (i) Let α1 , . . .Q , αk be distinct quantities. They are roots of f (x) if and only if f (x) = ki=1 (x−αi )g(x), where g(x) is a polynomial of degree n − k. (ii) The number of roots of f (x) is ≤ n. (iii) If f (x) is monic (meaning that a0 = 1) of degree n and has n (distinct) roots then n

f (x) = x + a1 x

n−1

n Y + · · · + an−1 x + an ≡ (x − αi ). i=1

As discussed later in the course, we can always assume that f (x) of degree n has n roots (not necessarily distinct). For example, if the coefficients ai of f (x) are rational numbers, then the “fundamental theorem of algebra” implies that f (x) has n complex roots αi . The proof of the fundamental theorem is analytic, and is given in topology (winding number) or in complex analysis (contour integral). 1

“Quantity” is explained in Exercise 2.3 below. For the moment, bear in mind the important special case ai ∈ Q and α ∈ C.

4

1.4

Relation between coefficients and roots

This section generalises the relations (1.3). Suppose given n quantities α1 , . . . , αn . We eventually intend them as the n roots of a polynomial f (x), but in this section we only treat them in formal identities, so that we could also think of them as independent indeterminates. Definition 1.3 The kth elementary symmetric function σk of the αi is defined by k X Y σk = αij . 1≤i1