Abstract. Let f (x) be a monic polynomial in Z [x] with no rational roots but with roots in Q p for all p, or equivalently, with roots mod n for all n. It is known that f (x) cannot be irreducible but can be a product of two or more irreducible polynomials, and that if f (x) is a product of m > 1 irreducible polynomials, then its Galois group must be a union of conjugates of m proper subgroups. We prove that for any m > 1, every finite solvable group that is a union of conjugates of m proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group of such a polynomial, and that the same result (with m = 2) holds for all Frobenius groups. It is also observed that every nonsolvable Frobenius group is realizable as the Galois group of a geometric, i.e. regular, extension of Q(t).

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.

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