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| Paper IPM / M / 18017 | ||||||||||||
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The aim of this paper is to consider Waring's problem for \SLn(D), where D is a division ring with center F such that dimFD≤4. We show that each element of \SLn(D) is a product of at most three square elements. As an application, let FG be a group algebra of a locally finite group G over a field F of characteristic p≠2. We show that if either p>2, or F is algebraically closed, or F is real-closed, or G is locally nilpotent, then every element in the derived subgroup (FG)′ is a product of at most three squares.
In this paper, we also discuss about decompositions of elements into products of torsion elements by showing, for instance, that if a field F contains at least n distinct torsion elements, then every element in SLn(F) is a product of at most two torsion elements.
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