Math 514a Algebraic Number Theory Fall 1995

Homework assignment 1: Read Chapter 1 of Hecke and write up solutions of the following problems. You should be ready to discuss them in class on Wednesday, August 30.

1. Check properties (i)-(v) on p. 6 of Hecke, for integers and for integer polynomials.

2. Prove that the set of residue classes of integers relatively prime to some fixed integer n forms a group under multiplication (which I denote (Z/nZ)^{×}). Use this to give another proof of Fermat’s little theorem (Theorem 9 of Hecke).

3. Prove the following generalization of Theorem 12 in Hecke: if R is an integral domain and f ∈ R[x] is a non-zero polynomial, then the number of roots of f in R is at most the degree of R.

4. Use the preceding exercise and the structure theorem for finite abelian groups to prove that a finite subgroup of the multiplicative group of a field is cyclic. In particular, (Z/pZ)^{×} is cyclic.

5. Use the last part of the preceding exercise to determine for which primes p the congruence

x^{2} + 1 ≡ 0 (mod p)

has a solution.