subfield

sub- +‎ field

subfield (plural subfields)

1. A smaller, more specialized area of study or occupation within a larger one
2. (algebra) A subring of a field, containing the multiplicative identity and closed under inversion.
   • 1953, Garrett Birkhoff with Saunders Mac Lane, A Survey Of Modern Algebra, Revised edition, U.S.A.: The Macmillan Company, published 1960, §XIV.1, page 394:

        Let us describe in general the subfield generated by a given element. Let K be a given field, F a subfield of K, and c an element of K. Consider those elements of K which are given by polynomial expressions of the form
     ${\displaystyle (1)\qquad f(c)=a_{0}+a_{1}c+a_{2}c^{2}+...+a_{n}c^{n}\qquad \qquad {\mbox{(each }}a_{i}{\mbox{ in }}F{\mbox{).}}}$
     [...]
        If f(c) and g(c) ≠ 0 are polynomial expressions like (1), their quotient f(c)/g(c) is an element of K, called a rational expression in c with coefficients in F. The set of all such quotients is a subfield; it is the field generated by F and c and is conventionally denoted by F(c), with round brackets.

   • 1953, Garrett Birkhoff with Saunders Mac Lane, A Survey Of Modern Algebra, Revised edition, U.S.A.: The Macmillan Company, published 1960, §XIV.2, page 397:

        We are now in a position to describe the subfield of K generated by F and our algebraic element u. This subfield F(u) clearly contains the subdomain F[u] of all elements expressible as polynomials f(u) with coefficients in F (cf. (1)). Actually, this domain F[u] is a subfield of K. Indeed, let us find an inverse for any element f(u) ≠ 0 in F[u].    [...]    This shows that F[u] is a subfield of K.
        Since, conversely, every subfield of K which contains F and u evidently contains every polynomial f(u) in F[u], we see that F[u] is the subfield of K generated by F and u.