subfield

Noun

subfield (plural subfields)

1. A smaller, more specialized area of study or occupation within a larger one
2. A subring of a field, containing the multiplicative identity and closed under inversion.
   • 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...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
   • 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...
   • 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.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