Galois Theory — Splitting Fields and Automorphisms

⚡ Key Takeaways

Splitting fields are the smallest field extensions in which a polynomial factors completely into linear factors; their automorphisms (field maps fixing the base) form the Galois group. One-line: for a separable, normal extension, $|\mathrm{Aut}(E/F)| = [E:F]$ , and the lattice of intermediate fields corresponds bijectively to subgroups of this group. Applies to: solvability of polynomials by radicals, constructibility, algebraic number theory, coding theory. Originated by Évariste Galois (1831), formalized by Artin (1940s).

💡 The Core Insight

The roots of an irreducible polynomial are interchangeable in a precise sense: any algebraic relation satisfied by one root with coefficients in $F$ is satisfied by all of them. The automorphisms of the splitting field permute the roots while preserving every such relation. Thus the Galois group is the group of "permutations of the roots that respect $F$ -arithmetic." Knowing this group is equivalent to knowing the entire web of intermediate fields — this is the Galois correspondence.

DefinitionSplitting Field

Let $F$ be a field and $f \in F[x]$ a nonconstant polynomial. An extension field $E \supseteq F$ is a splitting field of $f$ over $F$ if: 1. $f$ splits completely in $E[x]$ :

1

\quad f(x) = c\prod_{i=1}^{n}(x - \alpha_i), \qquad c \in F,\ \alpha_i \in E

Derivation hint: $f$ has degree $n$ ; in $E$ it acquires all $n$ roots (with multiplicity), so it factors into linear terms over $E$ . 2. $E$ is generated over $F$ by these roots:

2

\quad E = F(\alpha_1, \alpha_2, \ldots, \alpha_n)

Derivation hint: minimality — no proper subfield containing $F$ and all $\alpha_i$ exists; adjoining the roots generates exactly $E$ .

DefinitionAutomorphism Group of an Extension

For an extension $E/F$ , the group of $F$ -automorphisms is

3

\quad \mathrm{Aut}(E/F) = \{\sigma : E \to E \mid \sigma \text{ field iso},\ \sigma|_F = \mathrm{id}_F\}

Derivation hint: composition of field automorphisms fixing $F$ is again such; the identity and inverses also fix $F$ , giving a group under composition. When $E/F$ is finite, normal, and separable (a Galois extension), this group is denoted $\mathrm{Gal}(E/F)$ , the Galois group.

DefinitionNormal and Separable

$E/F$ is normal if every irreducible $p \in F[x]$ with a root in $E$ splits completely in $E$ . Equivalently, $E$ is the splitting field of some polynomial over $F$ . $E/F$ is separable if the minimal polynomial of every $\alpha \in E$ has no repeated roots. Over a field of characteristic 0 or a perfect field, every algebraic extension is separable.

TheoremExistence and Uniqueness of Splitting Fields

For any nonconstant $f \in F[x]$ , a splitting field $E$ exists and satisfies

4

\quad [E:F] \;\le\; (\deg f)!

Derivation hint: adjoin one root via $F[x]/(p(x))$ for an irreducible factor $p$ (degree $\le \deg f$ ), then induct on the remaining factor of degree one less; multiply the bounds. Moreover any two splitting fields of $f$ over $F$ are $F$ -isomorphic. The number of $F$ -automorphisms satisfies

5

\quad |\mathrm{Aut}(E/F)| \;\le\; [E:F],

with equality iff $E/F$ is separable and normal (i.e., Galois). Derivation hint: count extensions of an embedding root-by-root; each step multiplies by the number of distinct roots of the minimal polynomial, which equals the degree exactly when separable.

TheoremFundamental Theorem of Galois Theory

Let $E/F$ be a finite Galois extension with $G = \mathrm{Gal}(E/F)$ . There is an inclusion-reversing bijection between intermediate fields $K$ with $F \subseteq K \subseteq E$ and subgroups $H \le G$ :

6

\quad K \longmapsto \mathrm{Gal}(E/K), \qquad H \longmapsto E^{H} = \{x \in E : \sigma(x) = x\ \forall \sigma \in H\}

Derivation hint: $\mathrm{Gal}(E/K)$ collects automorphisms fixing the larger field $K$ , hence is a smaller subgroup; the fixed field $E^H$ grows as $H$ shrinks — both maps reverse inclusion and are mutually inverse by Artin's theorem. Degrees and indices match:

7

\quad [E:K] = |\mathrm{Gal}(E/K)|, \qquad [K:F] = [G : \mathrm{Gal}(E/K)]

Derivation hint: $E/K$ is itself Galois so equality holds in [5]; the tower law $[E:F]=[E:K][K:F]$ together with $|G|=[E:F]$ yields the index formula. The intermediate field $K$ is normal over $F$ iff $H = \mathrm{Gal}(E/K)$ is normal in $G$ , in which case

8

\quad \mathrm{Gal}(K/F) \;\cong\; G/H

Derivation hint: restriction $\sigma \mapsto \sigma|_K$ gives a homomorphism $G \to \mathrm{Gal}(K/F)$ when $K$ is $G$ -stable (normality); its kernel is exactly $H$ , and surjectivity follows from extending automorphisms.

ProofGalois Correspondence via Artin's Theorem

Step 1 — Fixed fields and degrees. Let $G$ be a finite group of automorphisms of a field $E$ , and set $F = E^G$ . Artin's theorem asserts

9

\quad [E : E^G] = |G|.

Derivation hint: the elements of $G$ are linearly independent over $E$ as functions (Dedekind's independence of characters); this bounds $[E:E^G]\ge|G|$ , and a counting of $E^G$ -linear relations gives the reverse bound.

Step 2 — The maps are well-defined and inverse. For $H \le G$ , clearly $H \subseteq \mathrm{Gal}(E/E^H)$ . By Step 1 applied to $H$ , $[E:E^H] = |H|$ , while $[E:E^H] = |\mathrm{Gal}(E/E^H)|$ since $E/E^H$ is Galois. Hence $|\mathrm{Gal}(E/E^H)| = |H|$ , forcing $H = \mathrm{Gal}(E/E^H)$ .

Step 3 — The other composite. For an intermediate field $K$ , $E/K$ is Galois (it remains separable and is the splitting field of the same $f$ now viewed over $K$ ). Thus $E^{\mathrm{Gal}(E/K)} = K$ — every element outside $K$ is moved by some $K$ -automorphism.

Step 4 — Normality and quotients. $K/F$ normal $\iff$ $\sigma(K) = K$ for all $\sigma \in G$ $\iff$ $\sigma H \sigma^{-1} = H$ for all $\sigma$ , i.e. $H \trianglelefteq G$ . The restriction map $G \to \mathrm{Gal}(K/F)$ then has kernel $H$ , and the first isomorphism theorem yields [8]. $\blacksquare$

∎

mathematician · Évariste Galois | 1811–1832

Galois, dead in a duel at twenty, wrote down on the eve of his death the memoir that founded the theory bearing his name. He introduced the word group and the idea that a polynomial's solvability is governed by the structure of a group of permutations of its roots. His manuscripts, rejected and lost by Cauchy and Fourier, were published only in 1846 by Liouville. The modern field-theoretic reformulation — splitting fields, automorphism groups, the correspondence — is due principally to Dedekind, Steinitz, and above all Emil Artin, whose 1942 lectures recast the whole subject around the fixed-field theorem rather than around resolvents.

🧠 Intuition

Picture the roots of an irreducible polynomial as a set of indistinguishable particles: from the vantage point of the base field $F$ , you cannot tell them apart. $\sqrt 2$ and $-\sqrt 2$ both satisfy $x^2-2=0$ , and no polynomial with rational coefficients distinguishes them — so the map $\sqrt 2 \mapsto -\sqrt 2$ is a symmetry of $\mathbb{Q}(\sqrt 2)$ . The Galois group catalogues all such consistent relabelings. Now stack symmetries. If you adjoin two independent square roots, $\mathbb{Q}(\sqrt 2, \sqrt 3)$ , you have two independent "flips," giving $\mathbb{Z}/2 \times \mathbb{Z}/2$ . Each subgroup corresponds to a sub-collection of symmetries you choose to keep fixed — and the field fixed by that subgroup is an intermediate field. The correspondence is a perfect dictionary: ask any question about which numbers lie in which subfield, and translate it into which automorphisms fix which elements.

Figure

The Galois correspondence for

\mathbb{Q}(\sqrt2,\sqrt3)/\mathbb{Q}

: subgroup lattice (left) mirrors field lattice (right), inclusion-reversed.

ExampleGalois group of

x^2-2

over

\mathbb{Q}

Given: $f(x) = x^2 - 2$ , irreducible over $\mathbb{Q}$ (no rational root).

Step 1 — Splitting field. Roots are $\pm\sqrt 2$ , so $E = \mathbb{Q}(\sqrt 2)$ . Degree:

10

\quad [E:\mathbb{Q}] = \deg(\text{min poly of }\sqrt2) = 2

Derivation hint: the minimal polynomial $x^2-2$ has degree 2, and $[\mathbb{Q}(\alpha):\mathbb{Q}]$ equals the degree of $\alpha$ 's minimal polynomial.

Step 2 — Automorphisms. Any $\sigma \in \mathrm{Gal}(E/\mathbb{Q})$ must send $\sqrt 2$ to a root of $x^2-2$ , i.e. $\sqrt2 \mapsto \pm\sqrt 2$ . Two choices, both valid (they preserve $(\sqrt2)^2=2$ ):

11

\quad \mathrm{id}:\ \sqrt2 \mapsto \sqrt2,\qquad \sigma:\ \sqrt2 \mapsto -\sqrt2

Derivation hint: an automorphism is determined by where it sends generators; it must permute roots of the minimal polynomial.

Step 3 — Group. $\sigma^2 = \mathrm{id}$ , so

\boxed{\mathrm{Gal}(\mathbb{Q}(\sqrt2)/\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z}, \quad |G| = 2 = [E:\mathbb{Q}]}

The only intermediate fields are $\mathbb{Q}$ and $\mathbb{Q}(\sqrt2)$ — matching the only subgroups $G$ and $\{e\}$ .

ExampleGalois group of

x^3-2

over

\mathbb{Q}

Given: $f(x) = x^3 - 2$ , irreducible by Eisenstein at $p=2$ .

Step 1 — Roots and splitting field. With $\omega = e^{2\pi i/3}$ a primitive cube root of unity, the roots are $\sqrt[3]2,\ \omega\sqrt[3]2,\ \omega^2\sqrt[3]2$ . So

12

\quad E = \mathbb{Q}(\sqrt[3]2,\ \omega), \qquad [E:\mathbb{Q}] = [E:\mathbb{Q}(\sqrt[3]2)]\,[\mathbb{Q}(\sqrt[3]2):\mathbb{Q}] = 2\cdot 3 = 6

Derivation hint: $[\mathbb{Q}(\sqrt[3]2):\mathbb{Q}]=3$ (cubic minimal poly); $\omega$ satisfies $x^2+x+1$ , degree 2, not contained in the real field $\mathbb{Q}(\sqrt[3]2)$ , so the tower multiplies to 6.

Step 2 — Automorphisms. $\sigma$ sends $\sqrt[3]2$ to any of three roots and $\omega$ to $\omega$ or $\omega^2$ : $3\times 2 = 6$ automorphisms, matching $[E:\mathbb{Q}]=6$ (so the extension is Galois). These permute the three roots of $f$ in all possible ways.

Step 3 — Group.

13

\quad \mathrm{Gal}(E/\mathbb{Q}) \cong S_3

Derivation hint: 6 automorphisms acting faithfully on 3 roots embed into $S_3$ (order 6); equal cardinality forces an isomorphism.

\boxed{\mathrm{Gal}(\mathbb{Q}(\sqrt[3]2,\omega)/\mathbb{Q}) \cong S_3,\quad |G|=6}

The subgroups of $S_3$ (one of order 3, three of order 2, plus $\{e\}$ and $S_3$ ) correspond exactly to the intermediate fields: $\mathbb{Q}(\omega)$ (normal, index 3 $\to$ quotient $\mathbb{Z}/2$ ), and the three conjugate non-normal fields $\mathbb{Q}(\omega^k\sqrt[3]2)$ .

⚠️ Common mistake

The Galois group is not "the symmetric group on the roots" in general. It is only a subgroup of $S_n$ — precisely those root-permutations preserving all $F$ -algebraic relations. For $x^4-10x^2+1$ the four roots $\pm\sqrt2\pm\sqrt3$ satisfy relations (their pairwise products are rational), forcing the group to be $\mathbb{Z}/2\times\mathbb{Z}/2$ , not $S_4$ .

Normality is about the extension, not about being a "nice" field. $\mathbb{Q}(\sqrt[3]2)/\mathbb{Q}$ is not normal: it contains one root of $x^3-2$ but not the complex ones. Adjoining $\omega$ repairs this.

connection

Splitting fields and their automorphisms sit at the crossroads of several theories. The Galois group encodes Solvability by Radicals: a polynomial is solvable by radicals iff its group is a Solvable Group, the obstruction for degree $\ge 5$ being the simplicity of $A_5$ . The correspondence is a finite-dimensional shadow of Infinite Galois Theory, where one equips the group with the Krull topology and restricts to closed subgroups. Over finite fields, every extension is Galois with cyclic group generated by the Frobenius Endomorphism $x\mapsto x^p$ , giving $\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)\cong\mathbb{Z}/n$ . The whole apparatus generalizes to Galois Cohomology and, via Grothendieck's Galois Theory, to a fundamental-group analogy linking field extensions and covering spaces. The structure of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ is the central object of modern Class Field Theory and the Langlands Program.

🔬 Where this appears

⏳ History

- 1770 — Lagrange studies resolvents and permutations of roots, anticipating group ideas in Réflexions sur la résolution algébrique des équations.

- 1799/1824 — Ruffini & Abel prove the general quintic is unsolvable by radicals; Abel's 1824 proof is rigorous but lacks a structural explanation.

- 1830–1832 — Galois writes the decisive memoir, introducing groups and the solvability criterion; killed in a duel May 1832.

- 1846 — Liouville publishes Galois's papers in his Journal, finally making them known.

- 1870 — Jordan systematizes Galois theory in Traité des substitutions, the first textbook treatment.

- 1893–1910 — Dedekind & Weber, Steinitz reframe the theory abstractly in terms of fields; Steinitz's 1910 Algebraische Theorie der Körper establishes splitting fields and algebraic closure.

- 1928 — Krull extends the correspondence to infinite extensions using a topological (profinite) group.

- 1942 — Artin delivers his Notre Dame lectures, basing everything on linear independence of characters and the fixed-field theorem — the modern formulation taught today.

- 1954 onward — Inverse Galois Problem intensively studied (Shafarevich proves solvable groups occur over

\mathbb{Q}

, 1954); general case still open.

open-problem

The Inverse Galois Problem asks: is every finite group the Galois group of some Galois extension of $\mathbb{Q}$ ? Despite vast progress — all solvable groups (Shafarevich), all symmetric and alternating groups, 25 of the 26 sporadic simple groups, and many groups of Lie type are known to be realizable — the general problem remains open. Even harder is the regular inverse Galois problem over $\mathbb{Q}(t)$ and the structural determination of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ , whose abelianization is described by class field theory but whose full structure is wildly mysterious — Grothendieck's theory of dessins d'enfants and the conjectural action through $\widehat{GT}$ (the Grothendieck–Teichmüller group) are among the deepest threads of contemporary number theory. In positive characteristic, inseparability obstructs the clean count $|\mathrm{Aut}|=[E:F]$ ; one must work with separable closures and purely inseparable factors. The differential and difference analogues (Picard–Vessiot, Galois theory of $q$ -difference equations) face their own open structural questions.

Galois Theory — Splitting Fields and Automorphisms

Formal Statement

Why It Holds

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also