Embedding division algebras in crossed products (Q1922478)

Let \(D\) be a division algebra having center \(F\) which is finite dimensional over \(F\). If there exists a maximal subfield \(E\) of \(D\) which is Galois over \(F\), \(D\) is called a crossed product (or \(G\)-crossed product if \(G=\text{Gal}(E/F)\)). If no such \(E\) exists, \(D\) is called a noncrossed product. Let \(UD(F,n,r)\) be the generic division algebra of degree \(n>1\) generated by \(r\) \((n,n)\) generic matrices over \(F\). They are noncrossed product by S. Amitsur (1972). But ``most'' of them embedd into crossed products. Theorem 3. Let \(F\) be a field, \(n>1\). There exists an integer \(R\) such that if \(r>R\), then \(UD(F,n,r)\) centrally embeds in an \(S(n)\)-crossed product division algebra.