Question

1. A zero of a polynomial p(x) ∈ R[x] is an element α ∈ R such...

1. A zero of a polynomial p(x) ∈ R[x] is an element α ∈ R such that p(α) = 0. Prove or disprove: There exists a polynomial p(x) ∈ Z6[x] of degree n with more than n distinct zeros.

2. Consider the subgroup H = {1, 11} of U(20) = {1, 3, 7, 9, 11, 13, 17, 19}.

(a) List the (left) cosets of H in U(20)

(b) Why is H normal?

(c) Write the Cayley table for U(20)/H.

(d) Is U(20)/H isomorphic to Z4 or Z2 ⊕ Z2?

Number 1 and 2 please! Thank you

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