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Thread: Central simple algebra

  1. #1
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    Central simple algebra

    These two questions are the exercises in Hungerford's Algebra(p.462 ex.1 and ex.5).

    1. If A is afinite dimensional central simple algebra over the field K, then
    A tensor the opposit ring of A over K is isomorphic to a space of n by n
    matrix over K. Where n is the dimension of A over K.

    2. If A is a finite dimensional central simple algebra over a field K, then
    the dimension of A over K is a perfect square.

    Thank you!!!
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  2. #2
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    Quote Originally Posted by chipai View Post
    These two questions are the exercises in Hungerford's Algebra(p.462 ex.1 and ex.5).

    1. If A is afinite dimensional central simple algebra over the field K, then
    A tensor the opposit ring of A over K is isomorphic to a space of n by n
    matrix over K. Where n is the dimension of A over K.
    define the map $\displaystyle f: A \times A^{op} \longrightarrow \text{End}_K(A)$ by $\displaystyle f(a , b)(x)=axb,$ for all $\displaystyle a,b,x \in A.$ this map is K-bilinear and so it induces a K-module homomorphism $\displaystyle f: A \otimes_K A^{op} \longrightarrow \text{End}_K(A)$ defined by

    $\displaystyle f(a \otimes b)(x)=axb,$ for all $\displaystyle a,b,x \in A.$ this map now is also an algebra homomorphism because $\displaystyle f((a \otimes b)(c \otimes d))(x)=f(ac \otimes db)(x)=acxdb=f(a \otimes b)(cxd)=f(a \otimes b)f(c \otimes d)(x).$

    also $\displaystyle f$ is injective because $\displaystyle A \otimes_K A^{op}$ is a (central) simple algebra and so $\displaystyle \ker f,$ which is an ideal of $\displaystyle A \otimes_K A^{op},$ must be trivial. finally $\displaystyle \dim_K A \otimes_K A^{op} = n^2 = \dim_K \mathbb{M}_n(K)=\dim_K \text{End}_K (A)$

    and therefore $\displaystyle A \otimes_K A^{op} \cong \text{End}_K(A) \cong \mathbb{M}_n(K).$


    Note: some facts that are used here:

    1) $\displaystyle \text{End}_K (V) \cong \mathbb{M}_n(K),$ for any $\displaystyle n$ dimensional $\displaystyle K$ vector space $\displaystyle V.$ we know this from elementary linear algebra.

    2) the tensor product of a central simple algebra and a simple algebra is a simple algebra. also the tensor product of two central simple algebras is central simple.

    3) for any finite dimensional $\displaystyle K$ vector spaces $\displaystyle V,W$ we have $\displaystyle \dim_K V \otimes_K W = (\dim_K V)(\dim_K W).$

    finally note that $\displaystyle A^{op}$ cannot be replaced by $\displaystyle A$ because then $\displaystyle f$ would not be a ring homomorphism.


    2. If A is a finite dimensional central simple algebra over a field K, then
    the dimension of A over K is a perfect square.

    Thank you!!!
    first note that if $\displaystyle F/K$ is any field extension, then $\displaystyle A \otimes_K F$ is an $\displaystyle F$-algebra and $\displaystyle \dim_F A \otimes_K F=\dim_K A.$ now if we assume that $\displaystyle F$ is the algebraic closure of $\displaystyle K$ and put $\displaystyle B=A \otimes_K F,$ then

    $\displaystyle B$ is a finite dimensional simple algebra. thus $\displaystyle B$ is semisimple and so by Artin-Wedderburn we have $\displaystyle B \cong \mathbb{M}_n(D),$ for some finite dimensional $\displaystyle F$ division algebra $\displaystyle D$ and some positive integer $\displaystyle n.$

    since $\displaystyle F$ is algebraically closed and $\displaystyle D$ is algebraic over $\displaystyle F,$ we must have $\displaystyle D=F.$ thus $\displaystyle B \cong \mathbb{M}_n(F)$ and therefore $\displaystyle \dim_K A = \dim_F B = \dim_F \mathbb{M}_n(F)=n^2.$
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