Which of the following subsets in the set of polynomials R[x] is not closed for the operation of polynomial addition?(a) {p ∈ R[x] | p(0) = }(b) {p ∈ R[x] | p' = 0 }(c) {p ∈ R[x] | p(1) = 0 }(d) {p ∈ R[x] | p(1) = p(0) }(e) R2[x]

a. Find a nonconstant polynomial in Z4[ x ], if one exists, that is a unit. b. Find a nonconstant polynomial in Z3[ x ], if one exists, that is a unit. c. Prove or disprove that there exist nonconstant polynomials in Zp[ x ] that are units if p is prime.

Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.

Prove Theorem If and are relatively prime polynomials over the field and if in , then in .