Mathematics 366
40)
Homework (due Oct. 21)
A. Hulpke
Let R = Q[x] and A = ⟨x 2 + x + 1⟩ ⊲ R. Find an inverse to A + (x + 1) in R/A.
41) Let R be a ring and A ⊲ R. Show that R/A is commutative if and only if rs−sr ∈ A for all r, s ∈ R.
42)
43∗ )
Determine the maximal ideals of Z (Hint: We have shown that every ideal in Z is principal.)
Let R = Z[x] and I = ⟨3, x 2 + 1⟩ ⊲ R. Show that I is a maximal ideal.
44) (GAP) Let F = Z2 be the field with two elements. Show that the rings (actually, both are fields
– you don’t need to show this) F[x]/ ⟨x 3 + x + 1⟩ and F[x]/ ⟨x 3 + x 2 + 1⟩ are isomorphic.
Hint: In the second ring, find an element α such that α 3 + α + 1 = 0 and use this to find how to
map I + x from the first ring. Then extend in the obvious way to the whole ring. In this problem you
are not required to check that the whole addition/multiplication tables are mapped to each other
(which would be rather tedious), just test about 2 nontrivial sums and products each.
45) a) Let R be a ring in which every ideal is a principal ideal. (The integers Z and polynomails
over a field F[x] are examples of such a ring.) Let I = ⟨p⟩ ⊲ R an ideal and I ≤ J ⊲ R a larger ideal.
Show that J = ⟨q⟩ for an element q ∈ R such that p = a ⋅ q with a ∈ R. (I.e. q divides p.)
b) Now let R = F[x] for a field F and let p(x) ∈ R a polynomial that has no proper factors (i.e. factors
of degree ≥ 1. Factors of degree 0 are just coefficients and do not count). We call such a polynomial
irreducible. Show – using a) – that ⟨p(x)⟩ is a maximal ideal (and therefore also a prime ideal) of R.
c) Conclude that in the case under b), F[x]/ ⟨p(x)⟩ is a field. (The fields with 4 and 9 elements we
constructed earlier in the course are exactly of this form!)
Problems marked with a ∗ are bonus problems for extra credit.