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Definition : Let $A$ be an $F$-algebra and let $K$ be an extension field of $F$. The $K$-algebra $K \otimes_F A$ is called the scalar extension of $A$ to $K$. It will be denoted by $A_K$.

Definition :Let $A$ be a finite dimensional central simple F-algebra. An extension field $K$ of $F$ is said to be a splitting field for $A$ if $A_K \cong M_n(K)$ for some $n \in N$.

Corollary : Let $A$ be a finite dimensional central simple algebra. If $S$ is a subalgebra of $A$ that contains the unity of $A$, then $C_A (S) \otimes A^{o } \cong End _S (A)$.

Theorem : Let $D$ be a finite dimensional central division algebra. Every maximal subfield $K$ of $D$ is a splitting field for $D$.

Proof: If $a \in C_D(K)$, then $a$ and $K$ generate a field. Therefore $a \in K$ since $K$ is maximal. That is, $C_D(K) = K$. Similarly, $C_{D^{o}} (K) = K$. According corolly, therefore shows that $K \otimes_{F} D$ and $End _K (D^{o})$ are isomorphic as F-algebras. Since $D^{o}$ is finite dimensional over $K$, we have $End K (D^{o})\cong M_n(K)$.

so, my questions :

1: In theorem, why is $K(a)$ a field?

2: why $C_D(K) = K$and $C_{D^{o}} (K) = K$?

3: why $End K (D^{o})\cong M_n(K)$ ?

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I.

Recall what a field is: a field is an abelian group (addition) with a multiplication such that the field minus $0$ forms an abelian group, and satisfies some axioms like the distributivity.

Now $K(a)$ with addition is obviously an abelian group. Since $a$ commutes with $K$, $K(a)^\times$ with multiplication is also an abelian group. The conclusion follows.

II.

$C_D(K)$ already contains $K$ as $K$ is a field. The above shows that $C_D(K)\subseteq K$. Similarly for $C_{D^o}(K)$.

III.

By the corollary (take $A=D^o$ and $S=K$), $K\otimes_FD\cong End_K(D^o)$.

But what is $End_K(D^o)$? It consists of $K$-linear endomorphisms on $D^o$. If $D^o$ has dimension $n$ over $K$, then this is nothing other than $M_n(K)$.


Hope this answers your question.

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