Fields and Galois theory

前置知识

27 项必需的前置知识（一直追溯到底）

你将学会

• 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.

定义

• 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.

定理

• 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.

常见误解

• 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.

通向

可选发现

• 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

学习路径

• Undergraduate mathematics major
• Graduate preparation
• Algebra and number theory

延伸阅读

• Ian Stewart, Galois Theory
• Emil Artin, Galois Theory (1942)