Sentence:  We are now in a position to describe the <WSD>subfield</WSD> 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 <WSD>subfield</WSD> of "K". Indeed, let us find an inverse for any element "f"("u") &ne; 0 in "F"["u"].    [...]    This shows that "F"["u"] is a <WSD>subfield</WSD> 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 <WSD>subfield</WSD> of "K" generated by "F" and "u". 
Meaning of the word marked with <WSD> tags:  A subring of a field, containing the multiplicative identity and closed under inversion.