Skolem–Noether theorem

In ring theory, a branch of mathematics, the Skolem–Noether theorem characterizes the automorphisms of simple rings. It is a fundamental result in the theory of central simple algebras.

The theorem was first published by Thoralf Skolem in 1927 in his paper Zur Theorie der assoziativen Zahlensysteme (German: On the theory of associative number systems) and later rediscovered by Emmy Noether.

Statement

In a general formulation, let A and B be simple unitary rings, and let k be the centre of B. Notice that k is a field since given x nonzero in k, the simplicity of B implies that the nonzero two-sided ideal BxB = (x) is the whole of B, and hence that x is a unit. Suppose further that the dimension of B over k is finite, i.e. that B is a central simple algebra of finite dimension. Then given k-algebra homomorphisms

f, g : A → B,

there exists a unit b in B such that for all a in A

g(a) = b · f(a) · b⁻¹.

In particular, every automorphism of a central simple k-algebra is an inner automorphism.

Proof

First suppose $B = \operatorname{M}_n(k) = \operatorname{End}_k(k^n)$. Then f and g define the actions of A on $k^n$; let $V_f, V_g$ denote the A-modules thus obtained. Any two simple A-modules are isomorphic and $V_f, V_g$ are finite direct sums of simple A-modules. Since they have the same dimension, it follows that there is an isomorphism of A-modules $b: V_g \to V_f$. But such b must be an element of $\operatorname{M}_n(k) = B$. For the general case, note that $B \otimes B^{\text{op}}$ is a matrix algebra and thus by the first part this algebra has an element b such that

$(f \otimes 1)(a \otimes z) = b (g \otimes 1)(a \otimes z) b^{-1}$

for all $a \in A$ and $z \in B^{\text{op}}$. Taking $a = 1$, we find

$1 \otimes z = b (1\otimes z) b^{-1}$

for all z. That is to say, b is in $Z_{B \otimes B^{\text{op}}}(k \otimes B^{\text{op}}) = B \otimes k$ and so we can write $b = b' \otimes 1$. Taking $z = 1$ this time we find

$f(a)= b' g(a) {b'^{-1}}$,

which is what was sought.

References

• A discussion in Chapter IV of Milne, class field theory [1]