Chapter IX. Fields and Galois Theory

Abstract

This chapter develops some general theory for field extensions and then goes on to study Galois groups and their uses. More than half the chapter illustrates by example the power and usefulness of the theory of Galois groups. Prerequisite material from Chapter VIII consists of Sections 1–6 for Sections 1–13 of the present chapter, and it consists of all of Chapter VIII for Sections 14–17 of the present chapter.

Sections 1–2 introduce field extensions. These are inclusions of a base field in a larger field. The fundamental construction is of a simple extension, algebraic or transcendental, and the next construction is of a splitting field. An algebraic simple extension is made by adjoining a root of an irreducible polynomial over the base field, and a splitting field is made by adjoining all the roots of such a polynomial. For both constructions, there are existence and uniqueness theorems.

Section 3 classifies finite fields. For each integer $q$ that is a power of some prime number, there exists one and only one finite field of order $q$, up to isomorphism. One finite field is an extension of another, apart from isomorphisms, if and only if the order of the first field is a power of the order of the second field.

Section 4 concerns algebraic closure. Any field has an algebraic extension in which each nonconstant polynomial over the extension field has a root. Such a field exists and is unique up to isomorphism.

Section 5 applies the theory of Sections 1–2 to the problem of constructibility with straightedge and compass. First the problem is translated into the language of field theory. Then it is shown that three desired constructions from antiquity are impossible: “doubling a cube,” trisecting an arbitrary constructible angle, and “squaring a circle.” The full proof of the impossibility of squaring a circle uses the fact that $\pi$ is transcendental over the rationals, and the proof of this property of $\pi$ is deferred to Section 14. Section 5 concludes with a statement of the theorem of Gauss identifying integers $n$ such that a regular $n$-gon is constructible and with some preliminary steps toward its proof.

Sections 6–8 introduce Galois groups and develop their theory. The theory applies to a field extension with three properties—that it is finite-dimensional, separable, and normal. Such an extension is called a “finite Galois extension.” The Fundamental Theorem of Galois Theory says in this case that the intermediate extensions are in one-one correspondence with subgroups of the Galois group, and it gives formulas relating the corresponding intermediate fields and Galois subgroups.

Sections 9–11 give three standard initial applications of Galois groups. The first is to proving the theorem of Gauss about constructibility of regular $n$-gons, the second is to deriving the Fundamental Theorem of Algebra from the Intermediate Value Theorem, and the third is to proving the necessity of the condition of Abel and Galois for solvability of polynomial equations by radicals—that the Galois group of the splitting field of the polynomial have a composition series with abelian quotients.

Sections 12–13 begin to derive quantitative information, rather than qualitative information, from Galois groups. Section 12 shows how an appropriate Galois group points to the specific steps in the construction of a regular $n$-gon when the construction is possible. Section 13 introduces a tool known as Lagrange resolvents, a precursor of modern harmonic analysis. Lagrange resolvents are used first to show that Galois extensions in characteristic 0 with cyclic Galois group of prime order $p$ are simple extensions obtained by adjoining a $p^\mathrm{th}$ root, provided all the $p^\mathrm{th}$ roots of 1 lie in the base field. Lagrange resolvents and this theorem about cyclic Galois groups combine to yield a derivation of Cardan's formula for solving general cubic equations.

Section 14 begins the part of the chapter that depends on results in the later sections of Chapter VIII. Section 14 itself contains a proof that $\pi$ is transcendental; the proof is a nice illustration of the interplay of algebra and elementary real analysis.

Section 15 introduces the field polynomial of an element in a finite-dimensional extension field. The determinant and trace of this polynomial are called the norm and trace of the element. The section gives various formulas for the norm and trace, including formulas involving Galois groups. With these formulas in hand, the section concludes by completing the proof of Theorem 8.54 about extending Dedekind domains, part of the proof having been deferred from Section VIII.11.

Section 16 discusses how prime ideals split when one passes, for example, from the integers to the algebraic integers in a number field. The topic here was broached in the motivating examples for algebraic number theory and algebraic geometry as introduced in Section VIII.7, and it was the main topic of concern in that section. The present results put matters into a wider context.

Section 17 gives two tools that sometimes help in identifying Galois groups, particularly of splitting fields of monic polynomials with integer coefficients. One tool uses the discriminant of the polynomial. The other uses reduction of the coefficients modulo various primes.