The Fundamental Theorem of Galois Theory

A finite extension $L/K$ is Galois when it is normal and separable, equivalently when the fixed field of $\mathrm{Aut}(L/K)$ is exactly $K$, equivalently when $|\mathrm{Aut}(L/K)|=[L:K]$. The Fundamental Theorem asserts that the maps $M\mapsto \mathrm{Gal}(L/M)$ and $H\mapsto L^H$ (the fixed field of $H$) are mutually inverse, inclusion-reversing bijections between intermediate fields $K\subseteq M\subseteq L$ and subgroups $H\le \mathrm{Gal}(L/K)$. This dictionary translates field-theoretic questions — which are continuous, geometric, and hard — into group-theoretic questions, which are finite, combinatorial, and often tractable. The surprising fact is its rigidity: normal subgroups correspond exactly to those intermediate fields that are themselves Galois over $K$, and the corresponding quotient group is the Galois group of that sub-extension. This single correspondence powers the classical proof that the general quintic is not solvable by radicals.

The Core Insight

Symmetry organizes structure. The automorphisms of $L$ fixing $K$ form a group whose subgroup lattice is a faithful, inverted photograph of the field lattice. Bigger groups fix smaller fields; bigger fields are fixed by smaller groups. The dimension counts and the symmetry counts coincide exactly ($|\mathrm{Gal}|=[L:K]$) only in the Galois case — that exact equality is what makes the correspondence a bijection rather than a mere injection.

Formal Statement

Galois Extension and Galois Group

Let $L/K$ be a field extension.

• $\mathrm{Aut}(L/K)$ is the group of field automorphisms $\sigma :L\to L$ with $\sigma {|}_K={\mathrm{id}}_K$, under composition.
• For a subset $S\subseteq \mathrm{Aut}(L/K)$, the fixed field is

• $L/K$ is finite Galois iff it is finite, normal (every irreducible $f\in K[x]$ with a root in $L$ splits in $L$), and separable (the minimal polynomial of every $\alpha \in L$ has no repeated roots). In that case we write $G=\mathrm{Gal}(L/K)=\mathrm{Aut}(L/K)$.

The defining numerical criterion (Artin) is:

Fundamental Theorem of Galois Theory

Let $L/K$ be a finite Galois extension with group $G=\mathrm{Gal}(L/K)$, $|G|=[L:K]=n$. Define

Then:

Why It Holds / Derivation

Proof of the Fundamental Theorem

The engine is Artin's Lemma: if $H$ is a finite group of automorphisms of a field $L$, then $[L:L^H]=|H|$.

Thus $\sigma (M)=M$ for all $\sigma$ (i.e. $M/K$ normal) iff $H$ is invariant under all conjugations, i.e. $H⊴G$.

Évariste Galois | 1811–1832

Galois, dead in a duel at twenty, never wrote "the Fundamental Theorem" — he wrote of the group of permutations of the roots that leaves all rational relations invariant. His 1831 memoir, rejected and nearly lost, contained the decisive idea: solvability by radicals corresponds to a chain of subgroups with abelian quotients (a solvable group). The modern lattice-theoretic statement, the fixed-field formalism, and the clean bijection are due to later hands — Dedekind introduced field automorphisms abstractly, Heinrich Weber systematized "Galois groups," and Emil Artin reorganized the entire theory around linear algebra and Artin's lemma, dethroning the primitive-element argument and making the proof characteristic-free and conceptually transparent.

Building Intuition

Think of $L$ as a crystal and $G$ as its symmetry group. Each intermediate field $M$ is a substructure you can carve out by demanding invariance under part of the symmetry. Demand invariance under the whole group $G$: you get the smallest possible field, $K$ (the most symmetric part). Demand invariance under only the trivial subgroup $\{1\}$: nothing is forced, you get all of $L$.

The order reversal is the natural "more constraints $\Rightarrow$ smaller object" logic: a larger subgroup imposes more symmetry constraints, so its fixed field is smaller. The degree of $L$ over $M$ literally counts how many symmetries you have left after pinning $M$ down — exactly $|\mathrm{Gal}(L/M)|$.

Normality is the subtle part. A subfield $M$ is "stable" — globally fixed as a set, not pointwise — exactly when the symmetries fixing it form a normal subgroup. Normal subgroups are the ones you can quotient by to get a new group; correspondingly normal subfields are the ones that carry their own self-contained Galois theory, with group the quotient.

The Galois correspondence for $\mathbb{Q}(\sqrt 2,\sqrt 3)/\mathbb{Q}$: fields (left) mirror subgroups (right)

Worked Examples

The biquadratic field $\mathbb{Q}(\sqrt2,\sqrt3)$

• $⟨{\tau }_3⟩$ fixes $\sqrt{2}$ ⟹ $L^{⟨{\tau }_3⟩}=\mathbb{Q}(\sqrt{2})$.
• $⟨{\tau }_2⟩$ fixes $\sqrt{3}$ ⟹ $\mathbb{Q}(\sqrt{3})$.
• $⟨{\tau }_2{\tau }_3⟩$ fixes $\sqrt{2}\sqrt{3}=\sqrt{6}$ ⟹ $\mathbb{Q}(\sqrt{6})$.

:::example[A non-abelian group: $\mathbb{Q}(\sqrt[3]{2},\omega )$] Given. Splitting field $L$ of $x^3-2$ over $\mathbb{Q}$, where $\omega =e^{2\pi i/3}$. Roots: $\sqrt[3]{2},\omega \sqrt[3]{2},{\omega }^2\sqrt[3]{2}$.

• $A_3⊴S_3$ ⟹ its fixed field is normal over $\mathbb{Q}$. $A_3$ (order 3, index 2) ⟹ fixed field has degree 2: $L^{A_3}=\mathbb{Q}(\omega )$, which is Galois (cyclotomic) ✓, with $\mathrm{Gal}(\mathbb{Q}(\omega )/\mathbb{Q})\cong S_3/A_3\cong \mathbb{Z}/2$.
• The three order-2 subgroups are not normal ⟹ their fixed fields $\mathbb{Q}(\sqrt[3]{2}),\mathbb{Q}(\omega \sqrt[3]{2}),\mathbb{Q}({\omega }^2\sqrt[3]{2})$ are not Galois over $\mathbb{Q}$ (each degree 3, but $x^3-2$ does not split in any one of them). They are the three conjugates of each other, matching the three conjugate subgroups.

Common Misconceptions

Galois is not automatic

The bijection requires $L/K$ to be both normal and separable. Either failure breaks the theorem:

• Inseparable failure: $K={\mathbb{F}}_p(t)$, $L=K(t^{1/p})$. Then $x^p-t=(x-t^{1/p}{)}^p$ has a single repeated root, $\mathrm{Aut}(L/K)=\{1\}$, yet $[L:K]=p$. The fixed field of the trivial group is all of $L$, not $K$, so [5] fails. The map $\Psi$ is not surjective onto $\{K\subseteq M\subseteq L\}$.
• Non-normal failure: $L=\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ has $\mathrm{Aut}=\{1\}$ but degree 3; $x^3-2$ does not split.

Order-reversal must not be dropped, and conjugates need not coincide

In $\mathbb{Q}(\sqrt[3]{2},\omega )$ the three subfields $\mathbb{Q}({\omega }^k\sqrt[3]{2})$ are distinct fields of the same degree 3 over $\mathbb{Q}$, corresponding to three distinct conjugate order-2 subgroups of $S_3$. A common error is to assume "same degree ⟹ same field" or "all index-3 subgroups are normal." Both are false here: the lattice genuinely has three separate branches, and none of these subfields is Galois over $\mathbb{Q}$ because the matching subgroups are not normal. Equation [9] is what forces conjugate subgroups to give conjugate (isomorphic but possibly unequal) subfields.

Infinite extensions need topology

For an infinite Galois extension (e.g. $\overline{\mathbb{Q}}/\mathbb{Q}$, or $\overline{{\mathbb{F}}_p}/{\mathbb{F}}_p$), the naive bijection fails: there are more subgroups than intermediate fields. The fix (Krull) is to topologize $G$ with the Krull (profinite) topology, where a base of open neighborhoods of $1$ is given by $\mathrm{Gal}(L/M)$ for finite $M/K$. Then the correspondence holds between intermediate fields and closed subgroups, not all subgroups. See Infinite Galois Theory.

Connections

The FTGT is a hub linking many fields:

• Solvability by Radicals: A polynomial is solvable by radicals iff its Galois group is a solvable group; the chain of subfields obtained by adjoining radicals mirrors a chain of normal subgroups with abelian quotients via [7].
• Class Field Theory: describes abelian extensions of number fields — those with abelian Galois group — via the idele class group; the FTGT's normality/quotient structure is the foundation.
• Inverse Galois Problem: which finite groups arise as $\mathrm{Gal}(L/\mathbb{Q})$? The correspondence makes this a precise question about realizing abstract groups.
• Covering Space Theory: the topological analogue — the deck transformation group of a covering corresponds to subgroups of the fundamental group, with normal subgroups giving regular (Galois) covers. This is the prototype of a Galois Category.
• Differential Galois Theory: replaces field automorphisms with those commuting with a derivation, yielding algebraic groups; solvability of linear ODEs in closed form mirrors solvability of the differential Galois group.

Real-World Applications

1. Insolvability of the quintic (Abel–Ruffini, via Galois). $\mathrm{Gal}$ of a generic quintic is $S_5$, which is not solvable; FTGT translates "no radical formula" into a group fact. The cornerstone of modern algebra curricula.
2. Ruler-and-compass impossibility. Doubling the cube and trisecting a general angle require degree-3 extensions whose Galois closures have non-2-power order; FTGT-based degree arguments settle 2000-year-old problems.
3. Coding theory (BCH and Reed–Solomon codes). Constructed over finite fields ${\mathbb{F}}_{q^m}$; the Frobenius generates the cyclic Galois group, and conjugate roots (Galois orbits) determine minimal polynomials of code generators.
4. Cryptography — elliptic curves and pairings. Frobenius eigenvalues, the Weil pairing, and Galois representations on torsion points $E[n]$ underpin ECC and isogeny-based post-quantum schemes.
5. Algebraic number theory — splitting of primes. The decomposition and inertia groups (subgroups of $\mathrm{Gal}$) govern how a prime factors in an extension; FTGT links these subgroups to the relevant subfields (decomposition/inertia fields).
6. Computer algebra systems. PARI/GP, Magma, and SageMath compute Galois groups to factor polynomials, identify number fields, and verify field isomorphisms — all leaning on the lattice correspondence.
7. Physics — anyons and topological order. Galois-theoretic constraints on modular tensor categories (Galois action on fusion/braiding data) classify admissible topological phases.

Historical Development

• 1770 — Lagrange. Réflexions sur la résolution algébrique des équations: studies how rational functions of roots permute, foreshadowing the "Galois resolvent."
• 1799–1813 — Ruffini. Argues (incompletely) that the general quintic is unsolvable by radicals.
• 1824 — Abel. Gives the first rigorous proof of quintic unsolvability, but without the conceptual group-theoretic framework.
• 1830–1832 — Galois. Introduces the group of an equation and the solvability criterion; memoir submitted to the Académie, rejected/lost. Dies in a duel May 1832 at 20.
• 1846 — Liouville. Publishes Galois's manuscripts in his Journal, finally making the work accessible.
• 1870 — Jordan. Traité des substitutions: first systematic treatment, framing Galois theory through permutation groups.
• 1893–1895 — Weber & Dedekind. Recast the theory abstractly using fields and automorphisms; "Galois group" enters standard language.
• 1928 — Krull. Extends the correspondence to infinite extensions via the profinite topology.
• 1942–1944 — Artin. Reformulates the entire subject around linear independence of characters and fixed fields (Artin's lemma), giving the clean, primitive-element-free proof used in modern graduate courses.

Subtleties & Limits

• Inverse Galois Problem. Is every finite group the Galois group of some extension of $\mathbb{Q}$? Known for all solvable groups (Shafarevich) and many simple groups, but open in general. The FTGT defines the question but does not solve it.
• Grothendieck's anabelian program. To what extent does the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, as a profinite group with its action on geometric fundamental groups, determine arithmetic varieties? The Section Conjecture remains open. This generalizes FTGT to a correspondence between covers and group actions (Grothendieck's Galois Theory).
• Explicit Langlands reciprocity. The Langlands program seeks to understand all (including non-abelian) Galois representations via automorphic forms — a vast generalization of the abelian case settled by class field theory; large parts are conjectural.
• Differential and difference Galois theory extend FTGT to linear ODEs/recurrences (Picard–Vessiot theory), with subtle constraints (algebraic closure of constants) and open inverse problems for differential Galois groups.

See Also