subfield

A subfield is a subset of a field that is itself a field under the same addition and multiplication as the larger field. Formally, if F is a field and K is a subset of F that contains 0 and 1 and is closed under addition, subtraction, multiplication, and taking multiplicative inverses of nonzero elements, then K is a subfield of F.

Every field contains a smallest subfield, called the prime subfield. If the field has characteristic 0, the prime subfield is isomorphic to the rational numbers, Q. If the field has prime characteristic p > 0, the prime subfield is isomorphic to the finite field Z/pZ. In this sense, Q or Z/pZ sits inside any field of characteristic 0 or p, respectively.

Common examples include the real numbers R and the complex numbers C, where Q is a subfield of both R and C. In fields of characteristic p, the finite field Z/pZ (often denoted GF(p)) embeds into the field as the prime subfield. Finite fields, or Galois fields, F_{p^n}, have subfields F_{p^d} for each d that divides n.

Subfields form a lattice under inclusion, reflecting how field extensions relate to each other. In Galois theory, the correspondence between subfields of a field extension and subgroups of the Galois group links the structure of subfields to group-theoretic symmetry. Identifying subfields is a central step in constructing field extensions, solving polynomial equations, and understanding algebraic structures.