Algebraic Number Theory
Math 514A Fall 2007
Problem Set 1
Due: Tuesday, Sep. 4th
1. Let K=Q[√d] be an imaginary quadratic field (i.e. d < 0 is a square free integer)
and NormK/Q:K→Qthe norm.
a) Prove or disprove: OKis Euclidean w.r.t. the norm iff d=−1,−2,−3,−7,−11.
b) Find the prime numbers which are of the form
p=x2
1+y2
1=x2
2+ 2y2
2=x2
3+x3y3+y2
3
with xi, yiintegers. How many such representations are there for a given p?
c) Describe the group of units O∗
Kof K. What is the maximal cardinality of
O∗
K?
2. Let K0=Q[√d0] and K00 =Q[√d00] be two quadratic number fields. Their com-
positum K=K0K00 is called a biquadratic number field.
a) Show that there exist relatively prime integers d0
0, d00
0such that K=Q[pd0
0,pd00
0].
b) Suppose that (d0, d00) = 1. Show that OK=OK0OK00 iff d0d00(d0−1)(d00 −1) = 0
(mod 8).
c) Find the rings of integers OKfor K=Q[√2,√3] and K=Q[√2,√−3].
3. In the polynomial ring A=Q[x, y], show that the principal ideal ℘= (x2−y3) is
a prime ideal, but the integral domain R=Q[x, y]/℘ is not integrally closed, and
find the integral closure of Rin the field of fractions of R.
4. Let K, L and Mbe number fields with K⊆L⊆M. Show for all α∈M,
T rL/K (T rM/L(α)) = T rM /K (α);
NL/K (NM/L(α)) = NM /K (α).
5. A number field of the form K=Q[α] with α3=da rational number is called
acubic field. In the case α3= 5, find the ring of integers OKand compute the
discriminant δK.
6. Stickelberger’s discriminant relation: The discriminant δKof an algebraic number
field Kis always ≡0 (mod 4) or ≡1 (mod 4).