Mathematics 466
Homework (due Sep. 19)
A. Hulpke
11) Show that in any ring we have that a(b − c) = ab − ac.
√
√
12) Show that Q[ 5] = {a + b 5 ∣ a, b ∈ Q} is a field.
13) Let R = Z3 [x] (i.e. the polynomials with coefficients taken modulo 3). Consider the polynomial f (x) = x 2 − 2.
a) We let S = {g ⋅ f ∣ g ∈ R} the set of all ring multiples of f . Show that S ≤ R.
b) Determine the cosets of S in R.
14)
Let R = {(
a b
a b
) ∣ a, b, c ∈ R} and φ∶ R → C, (
) ↦ a + bi. Show that φ is a ring
−b a
−b a
isomorphism.
15) a) Let R be a ring and S ≤ R a subring, and u ∈ R an arbitrary element. Let ϑ∶ S[x] → R,
∑ a i x i ↦ ∑ a i u i . Show that ϑ is a ring homomorphism.
i
i
√
b) Now let R = C, S = Q, and u = 3 2. Determine ker ϑ and Image(ϑ).
c) Show that for every u ∈ R
{ f (x) ∈ R[x] ∣ f (u) = 0} ⊲ R[x]
16) Get accustomed with the system GAP. (Nothing needs to be handed in.)