Finite semisimple group algebra of a normally monomial group

In this paper, the complete algebraic structure of the finite semisimple group algebra of a normally monomial group is described. The main result is illustrated by computing the explicit Wedderburn decomposition of finite semisimple group algebras of various normally monomial groups. The automorphism groups of these group algebras are also determined.

The Newton Polynomial

This chapter focuses on the Newton polynomial based on assumption that K is a differential-valued field of H-type with asymptotic integration and small derivation. Here K is also assumed to be equipped with a monomial group and (Γ‎, ψ‎) is the asymptotic couple of K. Throughout, P is an element of K{Y}superscript Not Equal To. The chapter first revisits the dominant part of P before discussing the elementary properties of the Newton polynomial. It then presents results about the shape of the Newton polynomial and considers realizations of three cuts in the value group Γ‎ of K. It also describes eventual equalizers, along with further consequences of ω‎-freeness and λ‎-freeness, the asymptotic equation over K, and some special H-fields.

On non-abelian Brumer and Brumer–Stark conjecture for monomial CM-extensions

Let K/k be a finite Galois CM-extension of number fields whose Galois group G is monomial and S a finite set of places of k. Then the "Stickelberger element" θK/k,S is defined. Concerning this element, Andreas Nickel formulated the non-abelian Brumer and Brumer–Stark conjectures and their "weak" versions. In this paper, when G is a monomial group, we prove that the weak non-abelian conjectures are reduced to the weak conjectures for abelian subextensions. We write D4p, Q2n+2 and A4 for the dihedral group of order 4p for any odd prime p, the generalized quaternion group of order 2n+2 for any natural number n and the alternating group on four letters respectively. Suppose that G is isomorphic to D4p, Q2n+2 or A4 × ℤ/2ℤ. Then we prove the l-parts of the weak non-abelian conjectures, where l = 2 in the quaternion case, and l is an arbitrary prime which does not split in ℚ(ζp) in the dihedral case and in ℚ(ζ3) in the alternating case. In particular, we do not exclude the 2-part of the conjectures and do not assume that S contains all finite places which ramify in K/k in contrast with Nickel's formulation.