Pries 605A Algebraic Number Theory spring 2016.
Homework 5. Due Fri 3/4.
1. Suppose K is a number field with r real embeddings and 2s complex embeddings. (So
r + 2s = [K : Q].) Show that dK has sign (−1)r .
√
2. Mimic
the
geometric
arguments
we
used
for
Z[
√ −5] to find a description of all ideals
√
in Z[ −6]. Find the ideal class group of Q( −6) using the definition.
3. Suppose I ⊂ OK is a non-zero ideal and a ∈ OK . If I m = (a) is principal for some
m ∈ N, show there exists α ∈ L = K(a1/m ) so that IOL = (α).
√
4. Use Minkowski’s bound to find the ideal class group of Q( 17).
Do one of these 3 problems (other 2 optional).
1. * Suppose K is a number field with |CLK | = 2. This problem shows that every
factorization of α ∈ OK into irreducibles has the same number of terms. Suppose π is
irreducible in OK but (π) is not a prime ideal of OK
A. If β is a prime ideal so that β|(π) show that β is not principal.
Q
B. Show (π) = ri=1 βi where βi are prime ideals (possibly equal) and r is even.
C. Show r = 2.
D. Suppose
K factors into irreducible elements of OK in two different ways:
Q α ∈ OQ
α = ti=1 πi = sj=1 λj . Show that t = s.
2. * The proof of Minkowski’s bound used two volume calculations. Prove these when
n = 3 (or for any n if you want):
(i) Vol(Xt ) = 2r−s π s tn /n!
p
(ii) Vol(Φ) = 2−s N (I) |dK |
√
3. * If K = Q( −14), show that the ideal class group of K is cyclic of order 4.