Computing Local Artin Maps, and Solvability of Norm Equations

Abstract

Let L = K(α) be an Abelian extension of degree n of a number field K, given by the minimal polynomial of α over K. We describe an algorithm for computing the local Artin map associated with the extension L/K at a finite or infinite prime v of K. We apply this algorithm to decide if a nonzero a ∈ K is a norm from L, assuming that L/K is cyclic.

A Database for Field Extensions of the Rationals

“…We have to determine whether K/M is embeddable into a Z 4 extension, which is the case if and only if −1 is a norm in K/M. This can be decided by applying the methods described in [1].…”

Algorithmic proof of the epsilon constant conjecture

“…The element σ ∈ G is a lift of the local norm residue symbol (p, F p /K p ) ∈ Gal(F p /K p ) ≃ Gal(F/K) with F being the maximal abelian subextension in L/K. An algorithm to compute local norm residue symbols is described in [1,Alg. 3.1].…”

A Framework for Approximating Qubit Unitaries

“…Condition (1) implies that group generated by elements from G is a subgroup of {U q : q ∈ M S }. Condition (2) can be checked using the framework developed in the recent paper [44] given set G M,S .…”

“…We will say that G M,S is a set of canonical generators of M S . Second, for each q from G Q find a representation of U q as a product of elements of G. For all q from G M,S we then can define: (1) Circuit(q) = (U 1 , . .…”

“…In practice, it is not necessary to check all primes of F , a finite set of primes is sufficient: as described in [1] the only primes that need to be checked are a) the divisors of the conductor F of K/F and b) all finite primes dividing the ideal eZ F . If e is benign, we therefore can efficiently compute the prime factorization and hence can perform this sufficient test for solvability of the norm equation.…”