The location of noncrossed products in Brauer groups of Laurent series fields over global fields

Since Amitsur’s discovery of noncrossed product division algebras in 1972, their existence over more familiar fields has been an object of investigation. Brussel’s work was a culmination of this effort, exhibiting noncrossed products over the rational function field k(t) and the Laurent series field k((t)) over any global field k—the smallest possible centers of noncrossed products. Witt’s theorem gives a transparent description of the Brauer group of k((t)) as the direct sum of the Brauer group of k and the character group of the absolute Galois group of k. We classify the Brauer classes over k((t)) containing noncrossed products by analyzing the fiber over χ for each character χ in Witt’s theorem. In this way, a picture of the partition of the Brauer group into crossed products/noncrossed products is obtained, which is in principle ruled by a relation between index and number of roots of unity. As a side consequence of the result there are crossed products that have a noncrossed product primary component.