subcampos

Subcampos, or subfields, of a field F are subsets S that are themselves fields under F's operations, containing 0 and 1 and closed under addition, subtraction, multiplication, and taking inverses of nonzero elements. The characteristic of a subcampo equals that of F, and every subcampo contains the prime field of F.

Examples include the rational numbers Q as a subcampo of the real numbers R and of the complex numbers C; in finite fields, F_p is a subcampo of F_{p^n} for any positive n; and the real algebraic numbers form a subcampo of R.

Subcampos are central to the study of field extensions and Galois theory. The degree [F:K] measures F as a vector field over a subcampo K. The subcampo generated by a subset A of F is the smallest subcampo containing A, obtained by closing A under the field operations. This generated subcampo contains all elements that can be formed from A using finitely many applications of addition, subtraction, multiplication, and taking inverses. Understanding subcampos helps describe how larger fields are built from smaller ones and how algebraic structures interact within a field.