(Dixmier) Let be a -algebra (in fact, any uncountable algebraically closed field will do) and be a simple -module having a countable basis as a -vector space. Then .

Brief outline:

- By Schur’s lemma, is a division algebra. Moreover, it has a countable basis.
- Suppose does not act as a scalar. Then is trascendental over .
- This is absurd since it would imply has uncountable dimension.

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In fact Quillen proved that this holds without cardinality assumptions on the field.

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