Field Extension

Why field extensions matter

Field extensions are the right framework for asking "does this equation have a solution?" — and once you have it, for asking "how does the answer relate to the question?" That second step turns into Galois theory and unlocks results no elementary technique reaches.

• Galois theory. The bridge between geometry of polynomial roots and abstract group theory. Solvability by radicals, ruler-and-compass constructions, and the impossibility of trisecting an arbitrary angle all reduce to facts about field-extension degrees and Galois groups.
• Algebraic number theory. Every number field K is a finite extension of ℚ. Its ring of integers ϴ_K, ideal class group, and unit group encode arithmetic information — Fermat's last theorem, the BSD conjecture, and class field theory all live here.
• Finite fields and coding theory. 𝔽_q with q = p^n is the unique field of order p^n up to isomorphism, and 𝔽_q/𝔽_p is a Galois extension of degree n with cyclic Galois group generated by Frobenius x ↦ x^p. Reed-Solomon codes, AES round constants, and elliptic-curve crypto all use this structure.
• Algebraic geometry. Function fields k(V) of varieties V are extensions of k. Generic point of V corresponds to k(V); transcendence degree of k(V)/k equals geometric dimension of V.
• Differential equations. Picard-Vessiot theory uses differential field extensions to ask when ODE solutions can be written in elementary form (Liouville's theorem on integration in elementary terms).
• Ruler-and-compass constructions. A length is constructible iff it lies in a tower of degree-2 extensions. Doubling the cube needs ∛2, but [ℚ(∛2):ℚ] = 3 is not a power of 2 — proves impossibility instantly.

Degree and the tower law

The single most-used technical tool in field theory is the tower law — for a chain K ⊆ L ⊆ M of fields, [M:K] = [M:L] · [L:K]. The proof is just bookkeeping: take a basis of M over L and a basis of L over K, multiply pairwise, get a basis of M over K. The consequences are deep.

• If [L:K] = n and α ∈ L, then deg(minimal polynomial of α over K) divides n.
• If α has minimal polynomial of degree d over K and β has minimal polynomial of degree e, then [K(α, β) : K] ≤ d · e, with equality iff the polynomials are "independent" in a precise sense.
• Constructible numbers form a subfield of ℝ that is a union of degree-2 extensions, so any constructible number has degree a power of 2 over ℚ.

Worked examples — minimal polynomials and degrees

Element α	Field K	Minimal polynomial	[K(α):K]
√2	ℚ	x² − 2	2
∛2	ℚ	x³ − 2	3
i	ℝ	x² + 1	2
e^(2πi/n) = ζ_n	ℚ	Φ_n(x), the n-th cyclotomic	φ(n)
√2 + √3	ℚ	x⁴ − 10x² + 1	4
π	ℚ	None — transcendental	∞
e	ℚ	None — transcendental	∞
α primitive of 𝔽_(p^n)	𝔽_p	Some irreducible of degree n	n

Splitting fields and normality

Given f(x) ∈ K[x], the splitting field of f over K is the smallest extension L containing all roots of f — equivalently, L = K(α₁, ..., α_d) where the α_i are the roots. Splitting fields are unique up to K-isomorphism.

An extension L/K is normal if it is the splitting field of some family of polynomials — equivalently, every irreducible polynomial in K[x] with one root in L has all its roots in L. ℚ(∛2)/ℚ is NOT normal (it contains one cube root of 2 but not the other two), while ℚ(∛2, ω)/ℚ is normal.

An extension is separable if the minimal polynomial of every element has no repeated roots. In characteristic 0 (and over finite fields) every extension is separable; in characteristic p, inseparable extensions exist (e.g. 𝔽_p(t)/𝔽_p(t^p) where x^p − t^p has only the root t with multiplicity p). A Galois extension is one that is both normal and separable.

The fundamental theorem of Galois theory

For a Galois extension L/K with Galois group G = Gal(L/K) of order [L:K], there is an inclusion-reversing bijection

• {intermediate fields K ⊆ M ⊆ L} ↔ {subgroups H ≤ G}
• M corresponds to H = Aut(L/M), the subgroup fixing M pointwise
• H corresponds to L^H = {x ∈ L : σ(x) = x for all σ ∈ H}, the fixed field
• [L:M] = |H| and [M:K] = [G:H]
• M/K is normal (a Galois extension) iff H is normal in G, in which case Gal(M/K) ≅ G/H

This dictionary turns "find all subfields" into "find all subgroups" — a finite-group computation. It is the engine behind Galois's solvability theorem and Wantzel's classification of constructible regular polygons.

Common misconceptions

• "ℚ(π) is a finite extension." π is transcendental over ℚ (Lindemann 1882), so [ℚ(π):ℚ] = ∞. ℚ(π) is isomorphic to the rational function field ℚ(x), not to any number field.
• "All extensions are Galois." Need both normal and separable. ℚ(∛2)/ℚ fails normality. 𝔽_p(t)/𝔽_p(t^p) fails separability. The Galois condition is what makes the fundamental theorem work.
• "Minimal polynomial is unique." Unique up to scaling; the minimal polynomial is the unique monic polynomial of smallest positive degree vanishing at α. Without monic, you have a one-parameter family of associates.
• "Adding one root adds all roots." Only true in characteristic 0 if the minimal polynomial has degree ≤ 2, or if the polynomial is x^n − a in a field already containing the n-th roots of unity. ℚ(∛2) misses two cube roots of 2.
• "[L:K] is the number of K-automorphisms." Equality holds only for Galois extensions. For ℚ(∛2)/ℚ, the degree is 3 but the only K-automorphism is the identity (the other two cube roots aren't in the field), so |Aut| = 1 ≠ 3.
• "Algebraic ⇒ finite degree." An algebraic extension can have infinite degree — the algebraic closure ℚ̄ of ℚ is algebraic but [ℚ̄:ℚ] = ∞ (it contains an n-th root of 2 for every n).