Feltutvidelser

Feltutvidelser (field extensions) describe a relation between two fields E and K in which K is a subfield of E. The pair is denoted E/K, and E is called an extension field of K. The degree [E:K] is the dimension of E as a vector space over K; when this dimension is finite, the extension is called finite; otherwise it is infinite.

An extension is algebraic if every element of E is a root of some nonzero polynomial with coefficients in K; otherwise it is transcendental. Among algebraic extensions, one distinguishes separable and inseparable extensions, depending on the roots of minimal polynomials in the given characteristic.

The tower law states that if K ⊆ L ⊆ E then [E:K] = [E:L]·[L:K], a relation that holds for finite extensions and, in the sense of dimensions, in many infinite cases as well.

If E/K is finite and separable, a primitive element theorem ensures E = K(α) for some α. In the finite case, many extensions are Galois: normal and separable. The Galois group Gal(E/K) consists of field automorphisms of E fixing K, and the Fundamental Theorem of Galois Theory provides a correspondence between intermediate fields and subgroups of Gal(E/K).

Common examples include ℚ ⊆ ℚ(√2) (degree 2), ℝ ⊆ ℂ (degree 2), and finite fields F_p ⊆ F_{p^n} of degree n. Extensions are central in number theory, algebraic geometry, and broader field theory.