Math 627
Homework #5
November 30, 2014
Due Tuesday, December 9
1. Let F be a finite field with q elements, let A = F [T ] be a polynomial ring in one variable,
and let A+ be the elements of A which are monic. We can define the Dedekind zeta
function for A as,
X 1
, s ∈ C, Re(s) > 1.
ζA (s) =
s
N
(a)
a⊆A
a6=(0)
(a) Show that ζA (s) =
X
a∈A+
1
q (deg a)s
.
1
.
1 − q 1−s
(c) For d ≥ 1, let ad denote the number of monic irreducible polynomials of degree
d. Show that
∞ −ad
Y
−ds
1−q
.
ζA (s) =
(b) If we group terms by degree, use the sum in (a) to show that ζA (s) =
d=1
−s
(d) If we let u = q , then combining (b) and (c), we have
∞
Y
−a
1
=
1 − ud d .
1 − qu d=1
Taking a logarithmic derivative with respect to u, prove that
∞
X
dad ud
qu
=
.
1 − qu
1 − ud
d=1
(e) Use (d) to prove that
X
dad = q n .
d|n
Then use Möbius inversion to prove
an =
1X
µ(d)q n/d .
n
d|n
√
2. Use√ the Class Number
Formula to find the class numbers of the fields Q( −7),
√
Q( −10), and Q( −23).
√
3. Use√the Class Number Formula to find the
√ class numbers
√ of the fields Q( 7) and
Q( 15). You may use the facts that 8 + 3 7 and 4 + 15 are fundamental units in
these fields.
1