Algebraic Number Theory
Math 514B Spring 2008
Problem Set 4
Due: Thursday, Feb. 21st
1. a) Find the discrimiannt of Q(ζ7+ζ−1
7)/Qin two ways: using derivatives, and
using group characters.
b) Fing the conductor of Q(ζ7+ζ−1
7)/Qin two ways: using the definition of
conductor, and using group charaters.
2. Fix an integer m > 1 and let s=σ+it be a complex number with σ > 1. For a
mod mDirichlet charater χ: (Z/mZ)∗→C∗, define the L-Series
L(s, χ) =
∞
X
n=1
χ(n)
ns=Y
p
1
1−χ(p)p−s.
a) Show that log L(s, χ) = −Pplog(1 −tp,s) in terms of some complex number
tp,s satisfying |tp,s|<1.
b) Show that log L(s, χ) = Ppχ(p)p−s+R(s), where |R(s)| ≤ 1
1−2−σPp
1
p2σ.
c) Conclude that we can write
log L(s, χ) = X
a∈(Z/mZ)∗
χ(a)( X
p≡a(m)
1
ps) + R(s) where |R(s)| ≤ π2
3.
3. Let K=Q(√−d) be an imaginary quadratic field, where d≡3 mod 4 is a rational
prime. Consider the L-Series L(s, χ) with the Dirichlet charater χ= ( −d
·) being
the Legendre symbol.
a) Show that ζK(s) = ζ(s)L(s, χ).
b) Find the relation between the value L(1, χ) and the class number hKof K.