1) If the rings R and S are isomorphic, show the rings R[x] and S[x] are isomorphic. 2) Let f(x) = 5x° + 3x3 + 1 and g(x) = 3x2 + 2x + 1 in Z Ax]. Determine the quotient and remainder when f(x) is...

1) If the rings R and S are isomorphic, show the rings R[x] and S[x] are isomorphic. 2) Let f(x) = 5x° + 3x3 + 1 and g(x) = 3x2 + 2x + 1 in Z Ax]. Determine the quotient and remainder when f(x) is divided by g(x). 3) Prove the Z [x] is not a principal ideal domain. 4) Show that an irreducible polynomial over R has degree one or degree two 5) Let f(x) = xs + 6 r Z Ax]. Write f(x) as a product of irreducible polynomials over Z 7. 6) Prove that the ideal is prime in Z [x] but not maximal in Z [x]. 7) Which of the numbers 3, 5, 7, 11, 13, 17, 19 are irreducible in the ring (?2)? 8) Let 17 denote the set of all elements of Z (?-5) of the form a + b?-5 with a = b(mod 2). Show that El is a maximal prime ideal of Z


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