Fields and Galois theory

Intuition Beginner

A field is a number system where you can add, subtract, multiply, and divide by anything nonzero. The rationals, the reals, and the complex numbers are fields; the integers are not (you cannot always divide).

A field extension is a larger field containing a smaller one — like the reals inside the complex numbers. The structure of an extension is measured by the symmetries that fix the smaller field and rearrange the larger one. Those symmetries form a group, the Galois group.

Galois theory is the dictionary between field extensions and their symmetry groups. It converts problems about solving equations into problems about groups, and it explains why no general formula exists for polynomial equations of degree five and higher.

Visual Beginner

A larger field drawn as a region containing a smaller field, with arrows showing the symmetries that fix the smaller field and permute the larger one.

The lattice of subfields on the left mirrors the lattice of subgroups on the right; this mirror is the Galois correspondence.

Worked example Beginner

The complex numbers contain the reals and add an element $i$ with $i\times i=-1$. The symmetry that fixes the reals must send $i$ to a root of $x\times x=-1$, which is either $i$ or $-i$.

Step 1. The symmetry fixes every real number.

Step 2. It sends $i$ to a root of the same equation. The two roots are $i$ and $-i$.

Step 3. So the Galois group of the complex numbers over the reals has two elements: the identity and complex conjugation $i\mapsto -i$.

What this tells us: the Galois group measures the symmetries of the extension, and for this extension it has exactly two elements.

Check your understanding Beginner

Formal definition Intermediate+

A field $F$ is a commutative ring with $1\ne 0$ in which every nonzero element has a multiplicative inverse. A field extension $K/F$ is an inclusion of fields $F\subset K$; the degree $[K:F]$ is the dimension of $K$ as a vector space over $F$ ^{[Lang §VIII]}.

The automorphism group $\mathrm{Aut}(K/F)$ consists of field automorphisms of $K$ fixing $F$ pointwise. The extension is Galois when $|\mathrm{Aut}(K/F)|=[K:F]$; in that case $\mathrm{Aut}(K/F)$ is the Galois group $\mathrm{Gal}(K/F)$.

For any intermediate field $F\subset L\subset K$, the fixed field of a subgroup $H\le \mathrm{Gal}(K/F)$ is $K^H=\{x\in K:\sigma (x)=x\text{ for all }\sigma \in H\}$.

Counterexamples to common slips

• Not every extension is Galois. $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ has degree 3 but automorphism group of order 1; it is not Galois.
• The Galois group is not the same as the automorphism group. The automorphism group can be smaller; Galois requires the extension to be normal and separable.
• Degree multiplies, but only for finite extensions. For towers $F\subset L\subset K$ with finite degrees, $[K:F]=[K:L][L:F]$.

Key theorem with proof Intermediate+

Theorem (Fundamental theorem of Galois theory). Let $K/F$ be a finite Galois extension with group $G=\mathrm{Gal}(K/F)$. The maps

$$L\mapsto \mathrm{Gal}(K/L),H\mapsto K^H$$

are mutually inverse, order-reversing bijections between intermediate fields $F\subset L\subset K$ and subgroups $H\le G$. Moreover, $L/F$ is Galois iff $\mathrm{Gal}(K/L)◃G$, in which case $\mathrm{Gal}(L/F)\cong G/\mathrm{Gal}(K/L)$.

Proof sketch. For an intermediate field $L$, the extension $K/L$ is Galois (since $K/F$ is), with group $\mathrm{Gal}(K/L)$. Artin's lemma gives $[K:K^H]=|H|$ for any finite subgroup $H\le \mathrm{Aut}(K)$. Combining: $K^{(\mathrm{Gal}(K/L))}=L$ (from Artin and degree counting) and $\mathrm{Gal}(K/K^H)=H$ (from finiteness). The order-reversal is immediate. The normal-subgroup statement follows because $L/F$ is normal iff conjugate fields coincide, which translates via the bijection to $H$ being normal. $\square$

Bridge. This theorem builds toward 21.05.01 (Galois representations), where the Galois groups of number-field extensions act on geometric objects and drive the Langlands program, and appears again in 03.12.02 (covering spaces), whose fundamental-group-to-covering correspondence is the topological analogue of the field-to-subgroup correspondence. The foundational reason Galois theory is a dictionary is that both fields and groups are encoded in the same lattice of intermediate structures, and putting these together, this is exactly the bridge that lets group-theoretic computations solve field-theoretic problems, and the pattern generalises to the covering-space and fundamental-group correspondences of algebraic topology.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none reflects that the project still needs a local bridge from Mathlib's Polynomial.SplittingField and GaloisGroup infrastructure to the downstream consumers (number-theoretic Galois representations in 21.05.01, arithmetic-geometric fundamental groups in 04.02.01). Mathlib itself has substantial field-and-Galois infrastructure.

Advanced results Master

The fundamental theorem converts field theory into group theory. Solvable groups correspond to extensions obtainable by a tower of radical adjunctions; the Abel-Ruffini theorem identifies the obstruction to solving a polynomial by radicals as the non-solvability of its Galois group. Cyclotomic fields $\mathbb{Q}({\zeta }_n)$, with Galois group $(\mathbb{Z}/n{)}^{\times }$, are the building blocks of abelian extensions of $\mathbb{Q}$, classified by Kronecker-Weber: every finite abelian extension of $\mathbb{Q}$ sits inside a cyclotomic field ^{[Jacobson Ch. 4]}.

Infinite Galois theory equips the Galois group with the Krull topology, under which the Galois correspondence becomes a bijection with closed subgroups. This is the bridge to étale fundamental groups in algebraic geometry 04.02.01 and to Galois representations in number theory 21.05.01: the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on every algebraic object defined over $\mathbb{Q}$, and its linear representations drive the Langlands program.

Synthesis. Galois theory is the load-bearing dictionary between field arithmetic and group theory across the curriculum: the fundamental theorem established here builds toward 21.05.01 where Galois groups act on geometric objects to produce Galois representations, appears again in 03.12.02 whose covering-space theory is the topological analogue of the Galois correspondence, the foundational reason the dictionary works is that both fields and groups are encoded in the same lattice of intermediate structures, the bridge is that the same lattice-reversal pattern recurs from subfields-and-subgroups to coverings-and-subgroups to submanifolds-and-subgroups in symmetric-space theory, and the pattern generalises to the differential Galois theory of differential equations and the Tannakian fundamental groups of tensor categories; putting these together, Galois theory is the central organising principle of algebra and the foundation on which number theory and arithmetic geometry build.

Full proof set Master

Proposition (Artin's lemma). Let $K$ be a field and $H$ a finite subgroup of $\mathrm{Aut}(K)$. Then $[K:K^H]=|H|$.

Proof sketch. The fixed field $K^H$ consists of elements fixed by every automorphism in $H$. For any $x\in K$, the orbit $\{h(x):h\in H\}$ has size dividing $|H|$, and the elementary symmetric functions of the orbit are $H$-invariant, hence lie in $K^H$. Thus every $x\in K$ satisfies a polynomial over $K^H$ of degree at most $|H|$, so $[K:K^H]\le |H|$. The reverse inequality uses Dedekind's lemma on the linear independence of distinct automorphisms: distinct automorphisms of $K$ over $K^H$ are linearly independent over $K$, so $|H|\le [K:K^H]$. Equality follows. $\square$

This lemma is the technical heart of the Galois correspondence: it ties the order of the automorphism group to the degree of the extension, making the lattice-reversal bijection work.

Connections Master

• Galois representations 21.05.01. The absolute Galois group of $\mathbb{Q}$ acts on every algebraic object over $\mathbb{Q}$; its linear representations are the central object of the Langlands program.

• Covering spaces 03.12.02. The Galois correspondence between subfields and subgroups mirrors the correspondence between covering spaces and subgroups of the fundamental group; the analogy is made precise by the étale fundamental group.

• Solvability and group theory 01.02.01. The classification of polynomials solvable by radicals reduces to the solvability of their Galois groups, driving the structural theory of finite groups.

• Algebraic geometry and fundamental groups 04.02.01. The étale fundamental group of a scheme generalises the absolute Galois group, with Grothendieck's Galois theory unifying the arithmetic and topological settings.

Historical & philosophical context Master

Évariste Galois developed the theory in 1830–1832, in the memoir Mémoire sur les conditions de résolubilité des équations par radicaux, written the night before the duel in which he died at age 20 ^{[Galois 1832]}. His insight was to recognise that the solvability of a polynomial equation is governed not by the size or form of its roots but by the symmetry group of its root system. This conceptual move — replacing equations by groups — founded modern algebra.

The theory was elaborated by Jordan (Traité des substitutions et des équations algébriques, 1870), Artin (whose 1942 lectures gave the modern abstract treatment via automorphism groups), and Grothendieck (whose SGA 1 developed the étale fundamental group as the geometric generalisation). The Krull-topology treatment of infinite Galois extensions came in the 1920s and underpins the modern arithmetic-geometric applications.

Galois theory is the conceptual foundation of the Langlands program, the largest active research program in number theory, which seeks to relate Galois representations to automorphic forms.

Bibliography Master

@book{LangAlgebra,
  author = {Lang, Serge},
  title = {Algebra},
  edition = {3},
  publisher = {Springer GTM 211},
  year = {2002},
}

@book{Jacobson1985,
  author = {Jacobson, Nathan},
  title = {Basic Algebra I},
  edition = {2},
  publisher = {W. H. Freeman},
  year = {1985},
}

@book{DummitFoote2004,
  author = {Dummit, David S. and Foote, Richard M.},
  title = {Abstract Algebra},
  edition = {3},
  publisher = {Wiley},
  year = {2004},
}

@misc{Galois1832,
  author = {Galois, Évariste},
  title = {Mémoire sur les conditions de résolubilité des équations par radicaux},
  year = {1832},
}