The discriminant of abelian number fields

Abstract

For an abelian number field K, the discriminant can be obtained from the conductor m of K, the degree of K over Q, and the degrees of extensions, where p runs through the set of primes that divide m, and p α is the greatest power that divides m. In this paper, we give a formula for computing the discriminant of any abelian number field. 2010 AMS Mathematics subject classification. Primary 11R18, 11R29. Keywords and phrases. Cyclotomic fields, abelian number fields, conductors, discriminant.The second author was supported by Consejo Nacional de Ciencia y Tecnología (CONACyT).

Rotated Z^n-Lattices via Real Subfields of Q(\zeta_2r)

“…Let L be the cyclotomic field Q(ζ 2 3 ) and K its maximal real subfield Q(ζ 2 3 + ζ −1 2 3 ). In this case, α = 2 − (ζ 3 2 3 + ζ −3 2 3 ), ∆ K = 2 3 and c = 2 2 . Considering the Z-basis {e 0 = 1, e 1 = e 1 = ζ 2 3 + ζ −1 2 3 } for O K and Q(x, y) = 1 2 2 q α (x, y) = 1 2 2 Tr K/Q (αxy), it follows that the matrix of q α is given by…”

Constructions of Dense Lattices of Full Diversity

“…If K = Q(ζ 9 + ζ −1 9 ), then [K : Q] = 3, {ζ 9 + ζ −1 9 , ζ 2 9 + ζ −2 9 , ζ 3 9 + ζ −3 9 } is an integral basis for K and d K = 3 4 . If M = {a 1 (ζ 9 + ζ −1 9 ) + a 2 (ζ 2 9 + ζ −2 9 ) + a 3 (ζ 3 9 + ζ −3 9 ) ∈ O K : 4a 1 + 4a 2 + a 3 ≡ 0(mod 6) and a 3 ≡ 0(mod 2)}, then M is a Z-submodule of O K of rank 3 and index 6.…”

“…Thus, t = min{Tr K/Q (α 2 ) : α ∈ M , α = 0} = 24, which is attained at a 0 = 1 and a 1 = a 2 = a 3 = 0. Since the volume of lattice σ (M ) equals 2 4 |d…”

“…Those parameters can be obtained in certain cases of lattices associated to number fields through algebraic properties [6]. From [4,Thorem 4.1], we can to give a lower (and upper) bound on the minimum product distance. The higher those two parameters are the more attractive the lattice becomes to be used for data transmission over Gaussian and fading channels.…”