General Directions: Written assignments should be submitted typeset. What you submit must represent your own work.
Decompose the polynomial into irreducible factors in the ring .
Decompose the polynomial into irreducible factors in the ring .
Let denote the ring of Gaussian integers.
Find the set of all units in .
Find a greatest common divisor in for and .
Let denote the ring . Explain why and have no greatest common divisor in . Hint: Look at the norms of these elements.
Let be an integer, and let denote the ring . Let denote the additive subgroup of given by
Find the smallest value of for which is an ideal in .
Find a familiar ring that is isomorphic to the quotient ring for the value of obtained in the previous part.