Skolem–Noether theorem

Remarks

The algebra $B \otimes_K A^{\mathrm{op}}$ is simple and finite-dimensional, so it has only one simple module up to isomorphism, and its modules are determined by their dimension. The maps $f$ and $g$ make $A$ into two $B \otimes_K A^{\mathrm{op}}$-modules, by $(b \otimes a) \cdot x = f(b) x a$ and $(b \otimes a) \cdot x = g(b) x a$, of the same dimension. An isomorphism $\varphi$ from the first to the second commutes with right multiplication by $A$, so $\varphi(x) = u x$ with $u = \varphi(1)$ invertible, and compatibility with $B$ gives $u f(b) = g(b) u$. Example: every automorphism of $M_n(K)$ as a $K$-algebra is conjugation $X \mapsto P X P^{-1}$ by an invertible matrix $P$. Example: any two elements of $\mathbb{H}$ with the same minimal polynomial $x^2 + 1$ over $\R$, such as $i$ and $j$, are conjugate in $\mathbb{H}$.

Going deeper

For a student

The theorem is used constantly in the theory of central simple algebras. It shows that $\Aut_K(M_n(K)) \cong \mathrm{PGL}_n(K)$, which identifies central simple algebras split by a Galois extension $L$ with classes in $H^1(\Gal(L/K), \mathrm{PGL}_n(L))$. Applied to a subfield $L$ of $A$ and its Galois conjugates, it is a key step in building the crossed product description of central simple algebras. Together with the double centralizer theorem it controls how subfields sit inside central simple algebras.