Galois theory connects field extensions to group theory, via the Galois group — the group of symmetries of a field extension. Its central achievement: a polynomial equation is solvable by radicals (has a formula like the quadratic formula) if and only if its Galois group is solvable. This resolved a question mathematicians had puzzled over since the 16th century: why is there no formula for the roots of a general degree-5 polynomial?
You probably know the quadratic formula: for ax² + bx + c = 0, x = (−b ± √(b²−4ac)) / 2a. There are similar (much messier) formulas for degree-3 and degree-4 equations. But mathematicians searched for centuries for a formula for degree-5 equations — and never found one. In 1830, a 19-year-old French mathematician named Évariste Galois figured out why: it's impossible. No such formula can exist.
Galois's insight was to associate each polynomial equation with a group — a set of symmetries describing how the roots of the equation can be permuted while preserving all their algebraic relationships. If this group has a certain kind of "solvable" structure, the equation can be solved with a formula. If not, it can't. For degree-5 (and higher) equations, the relevant group — the symmetric group S₅ — is not solvable, so no general formula exists.
Galois theory turned out to be far more than a tool for polynomial equations. It became a template for an entire way of thinking in mathematics: understanding an object by studying its symmetries. This idea now appears throughout modern mathematics and physics — including the classification of particles in the Standard Model.
A field extension K/F is a larger field K containing a smaller field F. For example, ℂ/ℝ is an extension: ℂ = ℝ(i), obtained by adjoining a root of x² + 1 = 0 to ℝ. The degree [K:F] is the dimension of K as a vector space over F. Here [ℂ:ℝ] = 2.
For a field extension K/F, the Galois group Gal(K/F) is the group of automorphisms of K that fix every element of F. An extension is a Galois extension if |Gal(K/F)| = [K:F] (the group is "as large as possible").
Example: for K = ℚ(√2, √3) over F = ℚ, the Galois group has 4 elements: identity, √2 → −√2, √3 → −√3, and both flipped simultaneously. This group is isomorphic to ℤ/2ℤ × ℤ/2ℤ.
For a finite Galois extension K/F with Galois group G:
This "Galois correspondence" is one of the most elegant results in all of algebra — it translates a hard field theory problem into an (often easier) group theory problem.
A polynomial is solvable by radicals if its roots can be expressed using the coefficients, the four arithmetic operations, and nth roots. Galois's Theorem: a polynomial is solvable by radicals if and only if its Galois group is a solvable group (a group with a chain of subgroups each normal in the next, with abelian quotients).
For degree ≤ 4, the relevant symmetric groups (S₂, S₃, S₄) are all solvable, which is why formulas exist. For degree ≥ 5, Sₙ contains the alternating group Aₙ, which is simple and non-abelian for n ≥ 5 — making Sₙ non-solvable. Hence no general radical formula exists for degree 5 or higher (the Abel–Ruffini theorem, first shown by Ruffini 1799 and Abel 1824, with Galois providing the complete conceptual explanation).
Galois submitted his memoir on the theory of equations to the French Academy of Sciences in 1831. Poisson, reviewing it, judged it "incomprehensible" and rejected it (a judgment historians now consider a failure to understand genuinely novel mathematics, though the manuscript was admittedly terse). Galois died in a duel on 30 May 1832, allegedly over a romantic dispute, though the exact circumstances remain historically murky. His friend Auguste Chevalier preserved his papers. Joseph Liouville, recognising their significance, published Galois's work in 1846 — 14 years after his death — in the Journal de Mathématiques Pures et Appliquées, finally making the theory available to the mathematical community.
Galois theory resolves three ancient Greek geometric construction problems, all asking whether certain constructions are possible using only a compass and unmarked straightedge:
The general principle: a length is constructible by compass and straightedge if and only if it lies in a field extension of ℚ obtained by a tower of degree-2 extensions — meaning [K:ℚ] must be a power of 2.
An open question: is every finite group realisable as the Galois group of some extension of ℚ? This is known for solvable groups (Shafarevich, 1954) and many specific simple groups, but remains open in general — one of the significant unsolved problems connecting group theory and number theory.
Grothendieck's approach to algebraic geometry generalises Galois theory to "Galois categories" and étale fundamental groups, extending the correspondence between symmetry groups and covering spaces far beyond classical field theory. This forms part of the conceptual foundation of modern arithmetic geometry, including work related to the Langlands program.