Galois theory answers a 300-year-old question — why is there no quadratic-formula analogue for degree-five polynomials? — by converting questions about equations into questions about symmetry groups. It is the capstone of the undergraduate algebra sequence and the archetype of a modern mathematical strategy: attach an algebraic invariant to a problem and let the invariant's structure decide what is possible.
A field extension K \subseteq L makes L a vector space over K; its dimension [L : K] is the degree. An element \alpha is algebraic over K if it satisfies a polynomial with coefficients in K; the smallest such monic polynomial (its minimal polynomial) has degree equal to [K(\alpha) : K]. Example: [\mathbb{Q}(\sqrt{2}) : \mathbb{Q}] = 2; [\mathbb{Q}(\sqrt[3]{2}) : \mathbb{Q}] = 3. The crucial bookkeeping fact is the tower law:
Degrees multiply — an innocent statement with lethal consequences below.
The splitting field of a polynomial f over K is the smallest extension where f factors completely into linear factors. The Galois group \mathrm{Gal}(L/K) is the group of field automorphisms of L fixing K pointwise. Each automorphism must permute the roots of f (it preserves every polynomial relation with coefficients in K), so the Galois group embeds into a permutation group of the roots — it is precisely the symmetries of the roots that respect all algebraic relations among them. Example: \mathrm{Gal}(\mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}) \cong \mathbb{Z}_2 \times \mathbb{Z}_2 — the four sign choices on the two square roots; the cyclotomic field of n-th roots of unity has Galois group (\mathbb{Z}/n\mathbb{Z})^\times, tying the subject to number theory.
For a Galois extension (normal + separable; over \mathbb{Q}, splitting fields of polynomials), the Galois correspondence is an inclusion-reversing bijection:
The entire lattice of intermediate fields — infinite-seeming field-theoretic data — is captured by the finite subgroup lattice of one finite group.
A polynomial is solvable by radicals if its roots can be written with arithmetic and n-th roots. Galois's theorem: this happens iff the Galois group is a solvable group — one admitting a chain of normal subgroups with abelian quotients (each radical adjunction corresponds to one abelian step). Degrees 2–4 have Galois groups inside S_2, S_3, S_4 — all solvable, hence the quadratic, cubic (Cardano), and quartic formulas. But S_5 is not solvable (its only proper normal subgroup A_5 is simple and non-abelian), and concrete quintics like x^5 - x - 1 have Galois group all of S_5. Therefore no general radical formula for degree ≥ 5 exists — not "none found": provably impossible, the impossibility residing in the structure of a 120-element group.
A constructible length lies in a tower of degree-2 extensions, so its degree over \mathbb{Q} is a power of 2 (tower law). Immediately: doubling the cube is impossible (\sqrt[3]{2} has degree 3), trisecting a general angle is impossible (\cos 20° has degree 3), and squaring the circle is impossible (\pi is transcendental — no finite degree at all). Three antique problems, closed by degree bookkeeping.
Galois theory is the template for modern mathematics' habit of studying objects through their symmetries: the same correspondence pattern reappears in covering spaces vs fundamental groups (topology), in differential Galois theory (which equations have closed-form antiderivatives — why e^{-x^2} has none is a Galois-style fact), and in the étale theory underlying modern number theory. Finite fields — whose extensions are the cleanest Galois theory of all, with cyclic Galois groups generated by the Frobenius map x \mapsto x^p — power coding theory and cryptography.