Math 772

Spring 2002

Exam 1

Due Wednesday February 20, 2002.

1. Show that the ring is a unique factorization domain (hint: use the norm to exhibit a Euclidean algorithm).
2. Consider the ring

a) Show that the smaller ring enjoys the property that given any element , there is a unit such that (so in a certain sense, every element of the larger ring is “almost” in the smaller ring).

b) Despite this unreasonable niceness, show that is not a unique factorization domain.

c) Show that is, however, an HFD (any two factorizations of the same nonzero nonunit have the same length).

3. Show that any positive prime (except 3) can be written in the form
4. For this problem let be the primitive 3^rd root of 1. We wish to find a formula for solutions to cubic equations. (This problem will illustrate the interplay of the roots of an equation and the role of symmetry).

a) Show that the equation can be translated to the equation by an appropriate change of variables.

b) Assume a solution of the form Show that the (reasonable?) assumption that the other two solution are of the form and its conjugate reduces the equation to a quadratic.

c) Derive the cubic formula (and astound yourself with some identities).