Math 103B
Test 1
100 pts
January 28, 2009
------------------------------------------------------------------------------------------------------Directions: Justify all answers and show all work.
Notation: Z is the ring of integers, Q is the field of rationals, M2(Q) is the ring of
2 by 2 matrices with rational entries, i = sqrt(-1).
Points: (1) 24 pts (2) 16 pts (3) 12 pts (4) 16 pts (5) 32 pts
-------------------------------------------------------------------------------------------------------(1) Briefly prove or disprove:
(A) Z[x] is an ideal of Q[x].
(B) M2(Q) is an integral domain.
(C) Z/4Z is an integral domain.
----------------------------------------(2) Prove that < x > is a maximal ideal of Q[x].
------------------------------------------------------------(3) Let S = {f(x) in Z[x] : f(i) is in 3Z }. (For example, x2 + 4 is in S .)
Prove that S is NOT an ideal of Z[x] .
------------------------------------------------------------------------------------------(4) Consider the ideal I = { f(x) in Z[x] : f(i) =0 } of Z[x] .
(For example, x2 + 1 is in I .) Prove that I is NOT maximal in Z[x] .
--------------------------------------------------------------------------------------------(5) Consider the quotient ring Z[i] / I, where I = < 3+3i > .
(A) Prove that the six cosets 0+I, 1+I, 2+I, 3+I, 4+I, 5+I are all distinct.
(B) The quotient ring has twelve other cosets besides those listed in (A).
List these remaining twelve cosets (no proof required).