Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: When is the multiplicative group of a field indecomposable?   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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The categories of this paper are: funct-an
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The categories of this paper are: eess
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The categories of this paper are: acc-phys
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The categories of this paper are: supr-con
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The categories of this paper are: hep-th
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: mtrl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: math-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin