Pries: 676 Number Theory. 2010. Homework 6.
Class group
g
1. Fix g > 1. Suppose that
√ n is odd, n > 1, and d = n − 1 is square-free. Show that the
ideal class group of Q( −d) has an element of order g.
2. Suppose I p ≡ J p in CLK and that p - |CLK |. Show that I ≡ J in CLK .
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. 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.
√
5. Use Minkowski’s bound to find the ideal class group of Q( 17).
6. * 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 |
√
7. * If K = Q( −14), show that the ideal class group of K is cyclic of order 4.