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1- What are the units in Z? What are the units in F[x]? Don’t write out a formal proof, but discuss why. 2- What is the analogy between Z and F[x]? 3- Let p(x) = x^3 + 3x + 1 = (x+3)^2 * (x+4) in Z5[x...

1- What are the units in Z? What are the units in F[x]? Don’t write out a formal proof, but discuss why.

2- What is the analogy between Z and F[x]?

3- Let p(x) = x^3 + 3x + 1 = (x+3)^2 * (x+4) in Z5[x].

(a) Perform the following computation in Z5[x]/(p(x)). Give your answers in the form [r(x)] where r(x) has degree as small as possible.

i. [4x] + [3x^2 + x + 2]

ii. [x^2][2x^2+1]

(b) Show that Z5[x]/((p(x)) has zero divisors.

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Answer #1

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1- What are the units in Z? What are the units in F[x]? Don’t write out a formal proof, but discuss why. 2- What is the analogy between Z and F[x]? 3- Let p(x) = x^3 + 3x + 1 = (x+3)^2 * (x+4) in Z5[x...
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