subfield

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 ).

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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.