How we proceed

f(1). Find a polynomial g(x) in Z[x] such that Ker d = (g(x)). Is there more than one possibility for g(x)? To what familiar ring is Z[x]/Ker o isomorphic? Do this exercise with Z replaced by Q. 37. Give an example of a field that properly contains the field of com- plex numbers C. 38. (Wilson's Theorem) For every integer n > 1, prove that (n - 1)! mod n = n - 1 if and only if n is prime. 39. For every prime p, show that (p - 2)! mod p = 1. 40. Find the remainder upon dividing 98! by 101. 41. Prove that (50!)2 mod 101 = -1 mod 101. 42. If I is an ideal of a ring R, prove that /[x] is an ideal of R[x]. 43. Give an example of a commutative ring R with unity and a maximal ideal I of R such that /[x] is not a maximal ideal of R[x]. 44. Let R be a commutative ring with unity. If I is a prime ideal of R, prove that /[x] is a prime ideal of R[x]. 45. Let F be a field, and let f(x) and g(x) belong to F[x]. If there is no polynomial of positive degree in F[x] that divides both f(x) and g(x) [in this case, f(x) and g(x) are said to be relatively prime], prove that there exist polynomials h(x) and k(x) in F[x] with the property that f(x)h(x) + g(x)k(x) = 1. (This exercise is referred to in Chapter 20.) 46. Prove that O[x]/(x2 - 2) is ring-isomorphic to @[V2] = la + b V 2 1 a , b E Q ) . 47. Let f(x) E R[x]. If f(a) = 0 and f'(a) = 0 [f'(a) is the derivative of f(x) at a], show that (x - a) divides f(x). 48. Let F be a field and let I = (f(x) E F[x] If(a) = 0 for all a in F}. Prove that I is an ideal in F[x]. Prove that I is infinite when F is fi- nite and I = {0} when F is infinite. When F is finite, find a monic polynomial g(x) such that I = (8(x)). 49. Let g(x) and h(x) belong to Z[x] and let h(x) be monic. If h(x) di- vides g(x) in O[x], show that h(x) divides g(x) in Z[x]. (This exer- cise is referred to in Chapter 33.) 50. Let R be a ring and x be an indeterminate. Prove that the rings R[x] and R[x ] are ring-isomorphic. 51. Let f(x) be a nonconstant element of Z[x]. Prove that f(x) takes on