Please do both problems. They are short.
I will rate for sure. Thanks!
Question# 1
Let p(x) = x^3 + 3x + 1 = (x + 3)^2 (x + 4) in Z5[x].
(a) Perform the following computations in Z5[x]/(p(x)). Giveyour answers in the form [r(x)] where r(x) has degree as small aspossible.
i. [4x] + [3x^2 + x + 2]
ii. [x^2 ][2x^2 + 1]
(b) Show that Z5[x]/(p(x)) has zero divisors
Question #2
Let p(x) = x^3 + x + 1 in Z2[x], and let R = Z2[x]/(p(x)).
(a) Explain briefly how you know that R has 8 elements.
(b) Is R is a field?
(c) Write out the multiplication table for R. (Don’t write outthe addition table.) I suggest that you omit the brackets.
