What does the degree of a field extension measure?

It is the dimension of the larger field as a vector space over the smaller one. A degree-two extension is obtained by adjoining a square root, for example, and degrees multiply when extensions are stacked into a tower.

How does this resolve angle trisection?

Constructible points generate extensions of degree a power of two. Trisecting a general angle would require solving an irreducible cubic, giving a degree-three extension, which is not a power of two, so it is impossible with straightedge and compass.

Field Extension

A field extension is a field containing a smaller field as a subfield, the basic object of field theory whose size is measured by its degree as a vector space.

Definition

A field extension is a pair consisting of a field and a subfield; equivalently, the larger field is regarded as a vector space over the smaller one, and the dimension of that vector space is the degree of the extension.

Scope

This topic covers the degree of an extension, algebraic versus transcendental elements, simple extensions and minimal polynomials, the tower law for degrees, finitely generated and algebraic extensions, and the application to classical straightedge-and-compass constructibility.

Core questions

How is the size of a field extension measured?
When is an element algebraic over the base field, and what is its minimal polynomial?
How do degrees multiply across a tower of extensions?
How does field theory resolve classical construction problems?

Key theories

Degree and the tower law: The degree of an extension is its dimension as a vector space over the base field, and in a tower of extensions the degrees multiply, making degree a fundamental additive-in-the-exponent invariant.
Minimal polynomial of an algebraic element: An element algebraic over a field is the root of a unique monic irreducible polynomial, the minimal polynomial, whose degree equals the degree of the simple extension it generates.
Constructibility: A length is constructible by straightedge and compass only if it lies in a tower of degree-two extensions, so the degree of the extension it generates must be a power of two, settling the impossibility of doubling the cube and trisecting a general angle.

Clinical relevance

Field extensions are the framework for studying roots of polynomials and for constructing new number systems, including the complex numbers, algebraic number fields, and finite fields. They turn the classical Greek construction problems into degree computations and underlie Galois theory.

History

Kronecker showed how to adjoin a root of a polynomial to a field by quotienting a polynomial ring, giving extensions an algebraic construction. Steinitz's 1910 memoir built the abstract theory of fields and their extensions, and Wantzel had earlier used degree arguments to prove the impossibility of several classical constructions.

Key figures

Leopold Kronecker
Ernst Steinitz
Évariste Galois
Pierre Wantzel

Seminal works

dummit2004
lang2002
artin2011

Frequently asked questions

What does the degree of a field extension measure?: It is the dimension of the larger field as a vector space over the smaller one. A degree-two extension is obtained by adjoining a square root, for example, and degrees multiply when extensions are stacked into a tower.
How does this resolve angle trisection?: Constructible points generate extensions of degree a power of two. Trisecting a general angle would require solving an irreducible cubic, giving a degree-three extension, which is not a power of two, so it is impossible with straightedge and compass.