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Math Help - Tensor product and Algebras

  1. #1
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    Tensor product and Algebras

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    Last edited by dabien; December 7th 2009 at 08:57 PM.
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  2. #2
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    Quote Originally Posted by dabien View Post
    Suppose we have finite field extensions B and K of F of respective degrees m and n, all contained in some larger field of E.

    Prove that the set of all finite linear combinations with coefficients in F of elements of E of the form \alpha*\gamma with \alpha \in B, \gamma \in K already form a subfield of E.
    what is * supposed to mean? is it the usual multiplication? if so, then write \alpha \gamma not \alpha * \gamma.
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  3. #3
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    i'll show the set of all finite linear combinations of the form \alpha \gamma, \ \alpha \in B, \ \gamma \in K with coefficients in F by Q. clearly Q is a subring of E and F \subseteq Q. also note that every element of Q is algebraic

    over F because B,K are algebraic over F. now let 0 \neq q \in Q. let f(x) =\sum_{i=0}^r a_ix^i \in F[x], \ a_r=1, be the minimal polynomial of q. so q^r + \cdots + a_1q + a_0 = 0. the assumption a_0 = 0 will contradict the minimality of f(x) because E is an integral domain. so a_0 \neq 0 and hence qa_0^{-1}(- q^{r-1} - \cdots - a_1)=1. therefore q is invertible in Q, because q^{-1}=a_0^{-1}(-q^{r-1} - \cdots - a_1) \in Q.
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