General Directions: Written assignments should be submitted typeset. What you submit must represent your own work.
Decompose the polynomial t^{12} - 1 into irreducible factors in the ring (Z/5Z)[t].
Decompose the polynomial t^{8} - 1 into irreducible factors in the ring (Z/2Z)[t].
Let G denote the ring Z + Z SQRT{-1} of Gaussian integers.
Find the set G^{*} of all units in G.
Find a greatest common divisor in G for 2 and 5 - SQRT{-1}.
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.
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}) . |
Find the smallest value of m >= 0 for which T_{m} is an ideal in R.
Find a familiar ring that is isomorphic to the quotient ring R/T_{m} for the value of m obtained in the previous part.