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arXiv · 1407.3481

When is the multiplicative group of a field indecomposable?

Sunil Chebolu·Keir Lockridge
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Abstract

The multiplicative group of a finite field is well known to be cyclic; in this note, we determine the finite fields whose multiplicative groups are direct sum indecomposable. We obtain our classification using a direct argument and also as a corollary to Catalan's Conjecture. Turning to infinite fields, we prove that any infinite field whose characteristic is not equal to 2 must have a decomposable multiplicative group. We conjecture that this is also true for infinite fields of characteristic 2 and we narrow the class of possible counter-examples. Finally, using the classification of finite commutative primary rings with cyclic multiplicative groups, we determine all finite commutative rings with indecomposable multiplicative groups.

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Sunil Chebolu, Keir Lockridge. 2014-07-13. When is the multiplicative group of a field indecomposable?. https://arxiv.org/abs/1407.3481

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