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?
Download

Mathematics 466 Homework (due Sep. 26) 17) 18)