Crossed Product Conditions for Central Simple Algebras in Terms of Splitting Fields

TL;DRAbstract

Let A be an F-central simple algebra of degree n (i.e. dimFA = n2). A criterion is given for A to be a crossed product simple algebra in terms of splitting fields of A. More precisely, it is shown that A is a crossed product if and only if there exists a Galois extension L/F such that [A] ∈ Br(L/F) and dimFL = mn with (m,n) = 1 and Gal(L/F) contains a normal subgroup of order m. Also, we prove that A1 ⊗F A2 is a nilpotent crossed product simple algebra if and only if A1 and A2 are nilpotent crossed product, where A1 and A2 are F-central simple algebras of relatively prime degrees m1 and m2.

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