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
(1) qquad f(c)=a_0+a_1c+a_2c²+...+a_ncⁿ qquad qquad mbox (eacha_i mbox inF 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.

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.

Subfield

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