I'm trying to follow the proof of $\mathrm{Br}(L/k)\simeq\mathrm{H}^{2}(L/k)=\mathrm{H}^{2}(\mathrm{Gal}(L/k), L^{\times})$ in Milne's Class Field Theory. In the proof, he use the following argument:
Let $f:A\to A'$ be an isomorphism between CSA over $k$ which have $L$ as a maximal subfield. Then by Noether-Skolem theorem, we can choose $f$ so that $f(L)=L$ and $f|_{L} = \mathrm{id}_{L}$. Hence....
I think this uses the following version of Noether-Skolem theorem:
Theorem 1. Let $A$ be a CSA over $k$ and $B_{1}, B_{2}$ are isomorphic simple $k$-subalgebras of $A$. Then the isomorphism can be extended to the inner automorphism of $A$.
However, in the book, Noether-Skolem theorem is stated as follows:
Theorem 2. Let $f,g:A\to B$ be $k$-algebra homomorphisms where $A$ is simple and $B$ is central simple over $k$. Then there exists $b\in B^{\times}$ s.t. $f(a) = b\cdot g(a)\cdot b^{-1}$ for all $a\in A$.
(I found the version 1 in the following paper.) I can't find how to prove Theorem 1 from the Theorem 2. Thanks in advance.
To prove Theorem 1, suppose $h:B_1\to B_2$ is an isomorphism. Let $g:B_1\to A$ be the inclusion and let $f:B_1\to A$ be the composition of $h$ with the inclusion $B_2\to A$. Applying Theorem 2 to $f$ and $g$, we get an element $b\in A$ such that $f(a)=bg(a)b^{-1}$ for all $a$. Plugging in the definitions of $f$ and $g$, this means $h(a)=bab^{-1}$, so $h$ extends to the inner automorphism given by conjugation by $b$.