Theorem 4

Let A be a central simple algebra of finite degree over K, and let L be a maximal commutative subalgebra of A. Then there exists a $\tau$-extension $\mathscr{E}$ of G by $L^*$ such that A is isomorphic to $\mathbf{A}[\mathscr{E}; L]$.

Without loss of generality, we may assume that L is a maximal commutative subalgebra of A. Let Γ be the multiplicative group consisting of the invertible elements $\gamma$ of A such that there exists a $g$ in G with

for every element $x$ of L. If $\gamma \in \Gamma$ is given, then the element $g$ satisfying this relation is unique; we denote it by $\pi (\gamma )$. It is immediate that $\pi$ is a homomorphism from Γ to G and that its kernel is equal to $L^*$. The surjectivity of $\pi$ follows from the Skolem–Noether theorem (VIII, p. 263, Corollary).

If we denote the canonical injection of $L^*$ into Γ by $\iota$, then it follows from the constructions that $\mathscr{E}= (\Gamma , \iota , \pi )$ is a $\tau$-extension of G by $L^*$. Let

$u: L\rightarrow \mathbf{A}[\mathscr{E}; L]$ and $v: \Gamma \rightarrow \mathbf{A}[\mathscr{E}; L]$

be the canonical homomorphisms. By the universal property of $\mathbf{A}[\mathscr{E}; L]$ (VIII, p. 315, Proposition 12), there exists a unique algebra homomorphism $f:\mathbf{A}[\mathscr{E}; L]\rightarrow A$ such that $f\circ u(x) =x$ and $f\circ v(\gamma ) =\gamma$ for $x\in L$ and $\gamma \in \Gamma$. Since the algebra $\mathbf{A}[\mathscr{E}; L]$ is simple, the homomorphism $f$ is injective. Now, the algebra $\mathbf{A}[\mathscr{E}; L]$ has degree $n^2$ over K and so does A because L is a maximal commutative semisimple subalgebra of A and we have $n= [L : K]$ (VIII, p. 262, Proposition 3, (ii)). Hence $f$ is bijective.