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MATH 3527 - Problem Set 7

1. (a) If (mod 4), prove that with the usual definitions of addition and multipli-cation, the set

fails to be a ring.

Hint: Find 2 elements of the set which when multiplied result in a complex number not included in the set.

(b) When (mod 4), the set with the usual notions of addition and multiplication is an integral domain. In this case, show that the set can be written as

Hint: Use the fact that is an integer polynomial having

as a root.

(c) Prove that the integral domain is a subring of .

2. (a) Suppose , a cube root of unity. Then the ring of Eisenstein integers is defined .  Show that any Eisenstein integer can be written in the form for integers a and b.

Hint: What quadratic polynomial with integer coefficients is a root of?

(b) In the Eisenstein integers, show that the usual notion of the norm map for a subring of C, the magnitude of a complex number squared, results in the formula

(c) Show that 13 is not prime in the Eisenstein integers by producing an explicit fac-torization into non-unit  elements.

3. Prove that a prime number p can be written as a sum of squares,

if and only if (mod 4) using the following steps:

(a) Show that p is not a prime element in .

Hint: Start by showing that p divides for some integer m using quadratic reciprocity.

(b) Deduce that p can be factored in into exactly 2 irreducible factors.

4. Show that is not prime in , but that is it irreducible.

5. (the Dirichlet ring) Consider the set of all arithmetic functions, R, with the oper-ations of convolution and point-wise addition, that is, for arithmetic functions f and g define a new arithmetic function (f + g)(n) = f(n) + g(n). This makes R a ring, called the Dirichlet ring.

(a) Prove that

(b) Prove that a f is a unit of R if and only if .

(c) Does the subset of multiplicative functions form a subring of R? Justify your an-swer.