advertisement

Mathematics 466 Homework (due Sep. 26) A. Hulpke 17) Show that 2Z and 3Z are not isomorphic. 18) In continuation of problem 13 we set R = Z3 [x], f (x) = x 2 − 2, and S = {g ⋅ f ∣ g ∈ R} ⊲ R. a) Let Q = R/S be the quotient ring. Construct (possibly using GAP) the addition and multiplication table. b) Show that the (classes represented by) polynomials of degree zero create a subring isomorphic to Z3 . (We shall identify Z3 with this subring. c) Show that the equation y 2 + 1 = 0 has no solution in Z3 . d) Show that the equation y 2 + 1 = 0 has a solution in Q. (We therefore could consider Q as Z3 [i].) e) Show that every nonzero element of Q has a multiplicative inverse. f) Now suppose we repeat the same construction, but with the polynomial x 2 − 1 = (x + 1)(x − 1). Do we get a field as well? Explain. 19) Let R be a ring and I, J ⊲ R be two ideals. a) Show that I ∩ J is an ideal. b) Let I + J = {i + j ∣ i ∈ I, j ∈ J} and IJ = {i ⋅ j ∣ i ∈ I, j ∈ J}. Show that both I + J and IJ are ideals of R. c) In R = Z, let I = ⟨4⟩, J = ⟨6⟩. Determine I + J, IJ and I ∩ J. 20) Let R = Q[x] and I = ⟨x 4 + 6x 3 + 8x 2 − 3x − 2, x 5 + x 4 − 7x 3 − 5x 2 + 7x + 3⟩ ⊲ R. a) Describe the elements of I in a way that makes it easy to test membership in I. Generalize this observation to describe all ideals in R. b) Determine an inverse of I + (x + 1) in R/I. 21) Let R be the ring of all functions defined on the set of real numbers. a) Let S be the set of all differentiable functions. Show that S ≤ R but S ⊲/ R. b) Let T = { f ∈ R ∣ f (1) = 0}. Show that T ⊲ R. c) Is T a principal ideal?