Written Assignment No. 5

due Wednesday, December 7, 2005

General Directions: Written assignments should be submitted typeset. What you submit must represent your own work.

Assigned Exercises

  1. Decompose the polynomial t^{12} - 1 into irreducible factors in the ring (Z/5Z)[t].

  2. Decompose the polynomial t^{8} - 1 into irreducible factors in the ring (Z/2Z)[t].

  3. Let G denote the ring Z + Z SQRT{-1} of Gaussian integers.

    1. Find the set G^{*} of all units in G.

    2. Find a greatest common divisor in G for 2 and 5 - SQRT{-1}.

  4. Let R denote the ring Z + Z SQRT{-5}. Explain why 14 and 6 + 2SQRT{-5} have no greatest common divisor in R. Hint: Look at the norms of these elements.

  5. Let m >= 0 be an integer, and let R denote the ring Z + Z SQRT{-5}. Let T_{m} denote the additive subgroup of R given by

     T_{m}  =  Z· 7 + Z·(m - SQRT{-5})    . 

    1. Find the smallest value of m >= 0 for which T_{m} is an ideal in R.

    2. Find a familiar ring that is isomorphic to the quotient ring R/T_{m} for the value of m obtained in the previous part.