The structure of the multiplicative group of residue classes modulo

Abstract

Let k be an algebraic number field of finite degree and be a prime ideal of k, lying above a rational prime p. We denote by G () the multiplicative group of residue classes modulo (N ≧ 0) which are relatively prime to . The structure of G () is well-known, when N = 0, or k is the rational number field Q. If k is a quadratic number field, then the direct decomposition of G () is determined by A. Ranum [6] and F.H-Koch [4] who gives a basis of a group of principal units in the local quadratic number field according to H. Hasse [2]. In [5, Theorem 6.2], W. Narkiewicz obtains necessary and sufficient conditions so that G () is cyclic, in connection with a group of units in the -adic completion of k.

A Possible Generalization of Maeda’s Conjecture

“…The group (O/m) * is an abelian group whose structure can be determined for example by the theorems in [Nak79]. These imply in particular that (Z/|D|Z) * can be seen as a subgroup of this group and, since it is abelian, as a quotient of it as well.…”

Faster and Lower Memory Scalar Multiplication on Supersingular Curves in Characteristic Three

“…× can also be derived, with some effort, from Nakagoshi's much more general results for generic unit groups of residue classes of orders of number fields modulo powers of arbitrary prime ideals [20] or from Halter-Koch's classification [16] for quadratic orders. However, the proof we give here is direct and much simpler, the expressions for the generators are explicit, and the generators also enjoy the property that they correspond to endomorphisms of E 3,µ that lend themselves to efficient evaluation.…”

The nonexistence of certain representations of the absolute Galois group of quadratic fields

“…First, we use the following theorem of Nakagoshi [12] to find a field A which contains all possible F 's.…”