Teilgebiet

Teilgebiet is a term used in algebra to denote a subfield. If F is a field, a Teilgebiet (or Unterkörper) K of F is a subset K ⊆ F that is itself a field under the same addition and multiplication as F, and contains the multiplicative identity 1 of F. Equivalently, K is closed under addition, additive inverses, multiplication, and taking inverses of nonzero elements, with the field operations restricted to K.

Every field F contains a smallest subfield, its prime field. If F has characteristic zero, this prime field is isomorphic to the rational numbers Q; if F has characteristic p > 0, it is isomorphic to the finite field F_p. The prime field sits inside F via the natural embedding of integers (or residues mod p).

Examples include Q as a subfield of R, and R as a subfield of C. For finite fields, if F = F_{p^n}, then F contains exactly one subfield of order p^d for each divisor d of n; in particular F_p ⊆ F_{p^n}.

If K is a Teilgebiet of F, then F is called an Erweiterungskörper (extension field) of K. The degree [F:K] measures the size of F as a vector field over K; in finite fields, the subfield structure is constrained by divisors of the extension degree.

Non-examples include subsets that fail to contain 1 or are not closed under the necessary operations; such sets are not subfields. Teilgebiete play a central role in field theory, Galois theory, and algebraic geometry by describing compatible arithmetic inside larger fields.