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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.)