Proof: Let A be the set of finite sums of elements of the form k1k2 for k1∈K1,k2∈K2, let φ:K1×K2→A be the bilinear map defined by φ(k1,k2)=k1k2, and let Φ:K1⊗K2→A be the corresponding F-linear transformation. We observeΦ(k1⊗k2)Φ(k′1⊗k′2)=k1k′1k2k′2=Φ((k1⊗k2)(k′1⊗k′2))allowing us to showΦ(∑iki1⊗ki2)Φ(∑jk′j1⊗k′j2)=∑i,jΦ(ki1⊗ki2)Φ(k′j1⊗k′j2)=∑i,jΦ((ki1⊗ki2)(k′j1⊗k′j2))=Φ((∑iki1⊗ki2)(∑jk′j1⊗k′j2))so that Φ is an F-algebra homomorphism. Note A=img Φ. As well, let K1 have for basis over F {ni} and let K2 have {mj}.
(⇒) We thus have Φ is a nonzero field homomorphism, and is thus an isomorphism. Now A is a field and by definition we have K1K2⊆A and by construction we observe A⊆K1K2 so that K1K2≅K1⊗K2 as F-algebras, the latter of which has for basis {ni⊗mj} of order [K1 : F][K2 : F]. (⇐) We still have A⊆K1K2, and since we observe [K1 : F][K2 : F] elements nimj of A linearly independent over F by Proposition 13.2.21, we must have this is a basis for K1K2 and thus again A=K1K2. The F-algebra homomorphism Φ above sends basis to basis and is thus an isomorphism, and now K1⊗K2 is a field. ◻
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