Galois theory · aiwiki.page

Galois theory is a branch of abstract algebra that studies field extensions through their symmetries. Its central result establishes a correspondence between intermediate fields and subgroups of an associated group, translating questions about algebraic quantities into questions in group theory. Originally developed to determine when polynomial equations can be solved by radicals, it also provides a framework for understanding the structure of algebraic extensions. (jmilne.org)

Historical development

The theory is named after Évariste Galois (1811–1832), whose work connected the solvability of equations with permutations of their roots. Rather than seeking a formula for every equation of a given degree, Galois investigated which symmetries permitted a particular equation to be solved by radicals. His mathematical papers were published posthumously by Joseph Liouville in 1846. (mathshistory.st-andrews.ac.uk)

Galois described groups using permutations that preserve algebraic relations among roots. The modern formulation instead uses fields and their automorphisms; these abstract structures were not available in their present form when he worked. (jmilne.org)

Fields, extensions, and symmetries

A field is a number-like algebraic structure in which addition, subtraction, multiplication, and division by nonzero elements are possible. An extension $L/K$ consists of a field $L$ containing a field $K$ . Its degree, $[L:K]$ , is the dimension of $L$ as a vector space over $K$ . An element is algebraic over $K$ if it satisfies a nonzero polynomial with coefficients in $K$ . (arxiv.org)

A field automorphism is a bijection from a field to itself preserving addition and multiplication. The automorphisms of $L$ fixing every element of $K$ form a group:

\operatorname{Aut}_K(L).

For a Galois extension, this is called its Galois group, denoted $\operatorname{Gal}(L/K)$ . (arxiv.org)

For a polynomial, the relevant extension is usually its splitting field: the smallest extension containing all its roots. Automorphisms fixing the coefficients permute these roots while preserving every polynomial relation among them over the base field. Thus the group records allowable algebraic symmetries, not arbitrary rearrangements. (jmilne.org)

Galois extensions and the fundamental theorem

An algebraic extension is Galois when it is both normal and separable:

Normality: every irreducible polynomial over the base field having a root in the extension splits completely there.
Separability: the minimal polynomial of each element has no repeated roots.

These hypotheses ensure that the automorphisms capture the extension’s structure sufficiently for the Galois correspondence. (arxiv.org)

Let $L/K$ be a finite Galois extension, with group $G$ . The fundamental theorem of Galois theory gives mutually inverse correspondences

E\longmapsto\operatorname{Gal}(L/E), \qquad H\longmapsto L^H,

where $K\subseteq E\subseteq L$ , $H\leq G$ , and

L^H=\{a\in L:\sigma(a)=a\text{ for every }\sigma\in H\}

is the fixed field of $H$ . The correspondence reverses inclusion: a larger intermediate field has a smaller group of automorphisms fixing it. (arxiv.org)

It also gives

|G|=[L:K],\qquad [L:E]=|H|,\qquad [E:K]=[G:H],

for $H=\operatorname{Gal}(L/E)$ . Moreover, $E/K$ is Galois precisely when $H$ is a normal subgroup of $G$ ; in that case,

\operatorname{Gal}(E/K)\cong G/H.

These statements allow subgroup structure to determine the degrees and relationships of intermediate fields. (math.mit.edu)

Example: the roots of a cubic

Consider $x^3-2$ over the rational numbers $\mathbb Q$ . Write $\alpha=\sqrt[3]{2}$ , and let $\omega$ be a primitive cube root of unity. The roots are

\alpha,\quad\omega\alpha,\quad\omega^2\alpha,

and the splitting field is

L=\mathbb Q(\alpha,\omega).

It has degree six over $\mathbb Q$ , and its Galois group is the symmetric group $S_3$ , acting by all six permutations of the roots. (kconrad.math.uconn.edu)

The subgroup structure of $S_3$ shows that $L$ has exactly four proper intermediate fields: the quadratic field $\mathbb Q(\omega)$ and the three cubic fields

\mathbb Q(\alpha),\quad \mathbb Q(\omega\alpha),\quad \mathbb Q(\omega^2\alpha).

The quadratic field corresponds to the normal subgroup of order three. The cubic fields correspond to nonnormal subgroups of order two, so they are not Galois over $\mathbb Q$ . In particular, adjoining one root need not produce a Galois extension. (kconrad.math.uconn.edu)

Solvability by radicals

A polynomial is solvable by radicals over $K$ if all its roots lie in an extension obtained through a finite sequence of adjunctions

K=K_0\subseteq K_1\subseteq\cdots\subseteq K_r, \qquad K_i=K_{i-1}(\beta_i),\quad \beta_i^{m_i}\in K_{i-1}.

This formalizes expressions built from field operations and extraction of roots. In characteristic zero, Galois’s criterion states that a polynomial is solvable by radicals if and only if its Galois group is solvable. (jmilne.org)

A solvable group admits a sequence of subgroups, each normal in the preceding one, whose successive quotient groups are abelian. This group-theoretic decomposition corresponds to a suitable sequence of simpler algebraic extensions. (math.bu.edu)

The groups $S_2,S_3,S_4$ are solvable, whereas $S_n$ is not solvable for $n\geq5$ . This explains the obstruction to general radical formulas beyond degree four. (math.bu.edu) For a concrete example, $x^5-x-1$ has Galois group $S_5$ over $\mathbb Q$ , so it is not solvable by radicals. This does not mean that every fifth-degree equation fails: $x^5-2$ , for example, has roots expressed using a fifth root of $2$ and fifth roots of unity. Nor does the restriction to radical formulas rule out numerical approximation. (math.uchicago.edu)

Geometric constructions and finite fields

In straightedge-and-compass construction, constructible coordinates lie in towers of quadratic extensions. Consequently, a constructible algebraic number has degree over $\mathbb Q$ equal to a power of two, although that degree condition alone is not sufficient. The cubic degree of $\sqrt[3]{2}$ therefore prevents the classical construction of a cube with twice the volume of a given cube. (arxiv.org)

For a finite field $\mathbb F_q$ , every finite extension $\mathbb F_{q^n}/\mathbb F_q$ is Galois. Its group is cyclic of order $n$ , generated by the Frobenius automorphism

a\longmapsto a^q.

For each divisor $d\mid n$ , there is exactly one intermediate field with $q^d$ elements. Thus finite fields provide an especially explicit realization of the Galois correspondence. (alozano.clas.uconn.edu)

Infinite extensions and scope

Galois theory also applies to infinite normal separable algebraic extensions. Their automorphism groups carry the Krull topology, obtained from restriction to finite Galois subextensions. The intermediate fields then correspond to closed subgroups, not to all subgroups indiscriminately. (jmilne.org)

With this topology, an infinite Galois group is a profinite group, assembled from finite groups through an inverse limit. Topology is therefore essential to the infinite correspondence rather than an optional addition. (arxiv.org)

The hypotheses also delimit the classical theory. Without normality or separability, the finite subgroup–field bijection need not hold. In positive characteristic, the characteristic-zero criterion for radical solvability cannot simply be transferred unchanged: separable equations of the form $x^p-x-a=0$ introduce a different kind of extension that must also be allowed in an appropriate solvability criterion. (arxiv.org)

References

Fields and Galois Theory — J. S. Milnejmilne.org
Fields and Galois Theoryjmilne.org
Évariste Galois — MacTutor History of Mathematicsmathshistory.st-andrews.ac.uk
Galois Theory — Tom Leinsterarxiv.org
A simple proof of the fundamental theorem of Galois theoryarxiv.org
Galois correspondence examples — Keith Conradkconrad.math.uconn.edu
704: Seminar in Algebra and Number Theorymath.mit.edu
Finite Fieldsalozano.clas.uconn.edu
Fields and Galois Theory — Infinite Galois Extensionsjmilne.org
Formalising the Krull Topology in Leanarxiv.org