Algebraic Number Theory
Math 514A Fall 2007
Problem Set 3
Due: Tuesday, Sep. 18th
1. Describe the decomposition/inertia groups of primes in the quadratic number field
K=Q[√19].
2. Describe the decomposition/inertia groups of primes in the cyclotomic field Q[ζ23 ].
3. Describe the decomposition/inertia groups of primes in the cyclotomic field Q[√2,√3].
4. Let pbe an odd prime number, ζa primitive pth root of unity. Let S=Pp−1
n=1(n
p)ζn
be the Gauss sum attached to the Legendre symbol. Show that S2= (−1
p)p. In
particular, if p∗= (−1
p)p, then √p∗∈Z[µp].
5. Define the Jacobi Symbol as follows: let m=p0p1. . . prand n=q1q2. . . qsbe
integers such that p0=±1 and pi, qjare prime numbers, nbeing odd. Suppose
that mand nare relatively prime. Define the Jacobi symbol by (m
n) = Qi,j(pi
qj),
where (pi
qj)0is the usual Legendre symbol. Prove the following:
a) If m1≡m2(mod n), then ( m1
n)0= (m2
n)0.
b) (m
n)0is multiplicative in both variables mand n.
c) One has (−1
n)0= (−1)(n−1)
2; Prove or disprove: (2
n)0= (−1)(n2−1)
8.
d) If mis odd too, then ( m
n)0= ( n
m)0(−1)(m−1)
2
(n−1)
2.
e) Prove or disprove: m(mod n) is a square in Z/n iff (m
n)0= 1
6. Consider the number field K0=Q[α], where α3= 2, and let K/Qbe its Galois
hull (i.e. the minimal extension of K0which is Galois over Q).
a) Show that K=Q[ζ3, α], where ζ3is a primitive 3rd root of 1, and that
G=Gal(K/Q)∼
=S3.
b) Detect the decomposition/inertia groups of prime ideals pof OKover rational
prime numbers p < 10.
c) Is there some psuch that its decomposition/inertia group is the whole Galois
group G?