Advanced undergraduateAlgebraPlanned

Fields and Galois theory

Field extensions and degrees, algebraic elements and minimal polynomials, splitting fields, finite fields, the fundamental theorem of Galois theory, and the unsolvability of the general quintic by radicals.

math.algebra.fields-galois

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Builds on

Quotient groups and quotient rings Linear algebra

Required prerequisites, all the way down: 27

You will learn to

Compute degrees of extensions with the tower law and find minimal polynomials.
Construct the finite fields 𝔽_{pⁿ} and prove that there is exactly one of each prime-power order up to isomorphism.
State and apply the fundamental theorem of Galois theory, and outline why the general quintic is not solvable by radicals.

Definition

The degree [L : K] of a field extension is the dimension of L as a vector space over K.
The Galois group Gal(L/K) is the group of field automorphisms of L that fix K pointwise.
A finite extension is Galois if it is normal and separable.

Theorem

Tower law: [M : K] = [M : L][L : K].
For a finite Galois extension L/K, intermediate fields correspond bijectively, reversing inclusion, to subgroups of Gal(L/K).
Abel–Ruffini: the general polynomial of degree n ≥ 5 is not solvable by radicals.
With straightedge and compass one cannot trisect a general angle or double the cube.

Common misconceptions

Reading Abel–Ruffini as saying that no quintic has roots expressible by radicals; specific quintics such as x⁵ − 2 do.
Confusing the ring ℤ/p²ℤ with the field of p² elements.

Leads to

Algebraic number theory

Optional discoveries

Adjunctions The Galois correspondence is an instance of a Galois connection, which is an adjunction between partially ordered sets (one taken with the opposite order).
← Complex numbers
← Polynomials
← Congruences
← Number theory