Definition 1

— Let K be a field. By an extension of K we understand a K-algebra whose underlying ring is a field. By a subextension (or sub-K-extension) of the extension E we understand a sub-K-algebra of E which is a field. Let E be an extension of K. The mapping u : A ↦ λ . 1 of K into E is a ring homomorphism; by I, p. 115, u induces an isomorphism of K onto a subfield u(K) of E. Conversely, if K, E are fields and u is a homomorphism of K into E, then u defines on E a structure of extension of K (III, p. 433). By abuse of language one sometimes says that (E, u) is an extension of K. An extension is said to be trivial if u(K) = E, that is, if E is a vector space of dimension 1 over K. Let L be an extension field of K. When we consider L as extension of K, we understand by this the extension (L, j) of K, where j is the canonical injection of K into L, or also L with the corresponding K-algebra structure. The subextensions of L are then the intermediate fields between K and L, that is, the subfields of L containing K. If L' is another extension field of K, a K-homomorphism of L into L' is then a homomorphism f of L into L' such that f(x) = x for all x ∈ K. We note that if f is any endomorphism of the field L, then the set of elements of L invariant under f is a subfield K' of L, and that f is then a K'-endomorphism of L. In particular let P be the prime subfield of a field L. We can consider L as an extension of P, and every endomorphism of L is then a P-endomorphism. Let (E, u) be an extension of K; since u defines an isomorphism of K onto a subfield K_1 of E, there is in general no difficulty in identifying K with K_1 by means of u. One case where we cannot allow such an identification is that where K = E and where u is thus an endomorphism of K; most frequently u will be an automorphism of K, or the mapping u ↦ u'', when the field K is of characteristic ≠ 0. It is clear that every extension of K is isomorphic to an extension (L, j), where L is a field containing K as subfield and j is the canonical injection of K into L.