Remark

— Let P be the prime subfield of a field E (V, p. 2); for every subset A of E, $P(A)$ is the least subfield of E containing A. In particular if K is a subfield of E, we have $P(K \cup A) = K(A)$. If K and K' are two subfields of E, we thus have $P(K \cup K') = K(K') = K'(K)$; this field is the least subfield of E containing K or K', or also the upper bound of K and K' in the set of subfields of E, ordered by inclusion; we sometimes say that this field is the field generated by K and K' in E.