Algebraic Number Theory
Math 514B Spring 2008
Problem Set 10 (The last one)
Due: Thursday, May 1st
1. Let L/K be a finite abelian extension, and {Hm(L/K)}={NL/K (Im
L)i(Km,1)}be
its class group.
a. Show that the class group is nonempty.
b. Show that for any representative Hmin the the class group {Hm(L/K)}, the
Artin map induces the isomorphism ϕL/K :Im
L
Hm∼
=Gal(L/K).
2. Fix positive integers mand n. Denote L=Q(ζm) and K=Q(ζn) as cyclotomic
fields, and m= (m)p∞and n= (n)p∞as moduli of Q.
a) Show that n|mif and only if Kis a subfield of L.
b) Say that ndivides m. Show that Hm=ImTi(Qn,1) is a congruence subgroup
mod m, and that the Artin map ϕL/Qinduces the isomorphism
Im
i(Qm,1)∼
=Gal(L/Q) while Im
Hm∼
=Gal(K/Q).
3. Fix a rational prime pand n, a divisor of p−1. Denote Las the unique subfield
of Q(ζp) of degree n,m= (p)p∞as a modulus of Q, and
Hm={(α)∈i(Qm)|α(p−1)/n ≡∗1 mod m}.
a) Show that Hmis a congruence subgroup modulo m.
b) Show that the Artin map induces the isomorphism Im/Hm∼
=Gal(L/Q).
4. Let Kbe a number field, and mbe a modulus of K. We call an extension Lof K
aray class field modulo mif
i. a prime of Kthat does not divide mis unramified in L, and
ii. the ray class group of Kis the Galois group of L/K, i.e.
ϕL/K :Im
i(Km,1)∼
=Gal(L/K).
Moreover, when mis the trivial modulus, we call LaHilbert class field.
a) Show that the ray class field, hence the Hilbert class field is unique.
b) Show that the Hilbert class field is always a subfield of the ray class field.
5. Fix pand qas distinct odd primes. Denote K=Q(√p∗q∗) and L=Q(√p∗,√q∗)
where p∗= (−1)(p−1)/2pand q∗= (−1)(q−1)/2q.
a) Show that no finite prime of Kramifies in L.
b) Show that the infinite primes ramify if and only if p=q= 3 mod 4.
c) Show that if not both pand qare congruent to 3 modulo 4, then Kdoes not
have class number 1.