Math 514a Algebraic Number Theory Fall 1995

Homework assignment 2: These problems are related to Chapters 1 and 3 of Hecke. You should also begin attacking Chapters 4 and 5 of Hecke and/or Chapter 2 of Marcus.

6. Prove that a positive integer n is prime if and only if n divides (n − 1)! + 1.

7. Prove that if F is a finite field, every element of F is the sum of two squares. (Hint: Count the images of the maps x ↦ x^{2} and y ↦ a − y^{2} for a fixed a ∈ F.) Is every element of F the sum of two cubes?

8. Is 91 a square modulo 2475? Is 2475 a square modulo 91?

9. Let p be a prime and m and a positive integers. True or False:

a) The equation x^{m} = a has a solution in Z/p^{n}Z for some n > 0.

b) The equation x^{m} = a has a solution in F_{p^{n}} for some n > 0. 10.

a) Prove that the congruence (x^{2} −13)(x^{2} −17)(x^{2} −221) ≡ 0 (mod m) has a solution for any m. (On the other hand, the equation (x^{2} − 13)(x^{2} − 17)(x^{2} − 221) = 0 obviously has no integer solutions.)

b) Fix integers a_{1}, a_{2} and b. Prove that the linear equation a_{1}x_{1} + a_{2}x_{2} = b is solvable in integers if and only if the congruence a_{1}x_{1} + a_{2}x_{2} ≡ b (mod n) is solvable for all n.