## Wedderburn’s factorization theorem (2)

Posted: October 13, 2011 in Division Rings, Noncommutative Ring Theory Notes
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You can see part (1) in here.

Wedderburn’s Factorization Theorem. (Wedderburn, 1920) Let $D$ be a division algebra with the center $k.$ Suppose that $a \in D$ is algebraic over $k$ and let $f(x) \in k[x]$ be the minimal polynomial of $a$ over $k.$ There exist non-zero elements $c_1, c_2 , \ldots , c_n \in D$ such that $f(x)=(x - c_1ac_1^{-1})(x - c_2ac_2^{-1}) \ldots (x - c_nac_n^{-1}).$

Proof. By Remark 2 in part (1), there exists $g(x) \in D[x]$ such that $f(x)=g(x)(x-a).$ Now let $m$ be the largest integer for which there exist non-zero elements $c_1, c_2 , \ldots , c_m \in D$ and $p(x) \in D[x]$ such that

$f(x)=p(x)(x - c_1ac_1^{-1})(x - c_2ac_2^{-1}) \ldots (x - c_m a c_m^{-1}).$

Let

$h(x) = (x - c_1ac_1^{-1})(x - c_2ac_2^{-1}) \ldots (x - c_mac_m^{-1}).$

Claim. $h(cac^{-1})=0$ for all $0 \neq c \in D.$

Proof of the claim. Suppose, to the contrary, that there exists $0 \neq c \in D$ such that $h(cac^{-1}) \neq 0.$ Then, since $f(x)=p(x)h(x)$ and $f(cac^{-1})=0,$ there exists $0 \neq b \in D$ such that $p(bcac^{-1}b^{-1})=0,$ by Lemma 1 in part (1). So, by Remark 2 in part (1), there exists $q(x) \in D[x]$ such that $p(x)=q(x)(x - bcac^{-1}b^{-1}).$ Hence $f(x)=p(x)h(x)=q(x)(x - bca (bc)^{-1})(x-c_1ac_1^{-1}) \ldots (x - c_m a c_m^{-1}),$ contradicting the maximality of $m. \ \Box$

Therefore, by Lemma 2 in part (1), $\deg h(x) \geq \deg f(x)$ and so $h(x)=f(x)$ because $f(x)=p(x)h(x)$ and both $f(x)$ and $h(x)$ are monic. $\Box$

In the next post, I will use Wedderburn’s factorization theorem to find an expression for the reduced trace and the reduced norm of an element in a finite dimensional central division algebra.