Algebraic Number Theory
Math 514B Spring 2008
Problem Set 2
Due: Tuesday, Feb. 5th
1. Fix an integer n > 1 along with a rational prime pnot dividing n. Denote R=Z[ζn]
as the integral closure of Zin K=Q(ζn), and pas a prime in Rlying over p.
Assume p≡1 mod n.
a) Show that there exists a homomorphism χ: (R/pR)∗→(Z/pZ)∗→C∗such
that χn= 1.
b) Show that the map ( ·
p)n: (R/p)∗→C∗defined by α(p−1)/n ≡(α
p)nmod pis
one such homomorphism. (This is the nth-power residue symbol.)
c) Denote the Gauss sum
S=X
α∈(R/p)∗
(α
p)n·ζa
p
and define a map χ:Gal(L/K)→C∗that sends σ7→ σ(S)
S. Show that χis
a homomorphism such that χn= 1.
d) Fix a prime q= 1 mod ndistinct from p, and choose αq∈R/pRcorrespond-
ing to q∈Z/pZ. Show that the Frobenius element corresponding to qmaps
to
χ(σq) = (αq
p)−1
n.
2. Let Kbe a real number field of degree n=r+ 2sin terms of the number rof
real places and the number sof pairs of complex places. We say Khas narrow
class number 1 if every nonzero fractional ideal is principal and is generated by a
totally positive number (i.e. elements α∈Ksuch that σi(α)>0 for the rreal
embeddings σi:K ,→R). Show that, with mbeing the modulus which is the
product of all of the infinite places of K, the following are equivalent:
a) Khas narrow class number 1;
b) Cm=Im/i(Km,1) is trivial;
c) The map Km/Km,1→Cmis surjective, and [UK:UKTKm,1] = 2r.
3. Let Kbe a number field of class number 1. Given a modulus m, let φ:Km→C∗
be a multiplicative character that is trivial on Km,1. Show that such a character
φfactors through a Dirichlet character mod m(i.e. φ=χ·iin terms of a multi-
plicative character χ:Im→C∗that is trivial on i(Km,1)) if and only if φis trivial
on UK.