Skolem–Noether theorem

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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 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 Bx 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. Then given k-algebra homomorphisms

f, g : AB

there exists a unit b in B such that for all a in A[1][2]

g(a) = b · f(a) · b−1.

In particular, every automorphism of a central simple k-algebra is an inner automorphism.[3][4]

Proof

First suppose . Then f and g define the actions of A on ; let denote the A-modules thus obtained. Any two simple A-modules are isomorphic and are finite direct sums of simple A-modules. Since they have the same dimension, it follows that there is an isomorphism of A-modules . But such b must be an element of . For the general case, note that is a matrix algebra and thus by the first part this algebra has an element b such that

for all and . Taking , we find

for all z. That is to say, b is in and so we can write . Taking this time we find

,

which is what was sought.

Notes

  1. Lorenz (2008) p.173
  2. {{#invoke:citation/CS1|citation |CitationClass=book }}
  3. Gille & Szamuely (2006) p.40
  4. Lorenz (2008) p.174

References

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