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Thursday, September 26, 2013

Tensor Products and Field Extensions (13.3.22)

Dummit and Foote Abstract Algebra, section 12.3, exercise 22:

MathJax TeX Test Page Let K/F be a finite field extension, and let K1,K2K be field extensions of F. Show the F-algebra K1FK2 is a field if and only if [K1K2 : F]=[K1 : F][K2 : F].

Proof: Let A be the set of finite sums of elements of the form k1k2 for k1K1,k2K2, let φ:K1×K2A be the bilinear map defined by φ(k1,k2)=k1k2, and let Φ:K1K2A be the corresponding F-linear transformation. We observeΦ(k1k2)Φ(k1k2)=k1k1k2k2=Φ((k1k2)(k1k2))allowing us to showΦ(iki1ki2)Φ(jkj1kj2)=i,jΦ(ki1ki2)Φ(kj1kj2)=i,jΦ((ki1ki2)(kj1kj2))=Φ((iki1ki2)(jkj1kj2))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 K1K2A and by construction we observe AK1K2 so that K1K2K1K2 as F-algebras, the latter of which has for basis {nimj} of order [K1 : F][K2 : F]. () We still have AK1K2, 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 K1K2 is a field. 

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