Use the method outlined in remark 2 to compute equation of

Question

Algebraic Number Theory Questions -

Q1. Let a be an ideal of a ring R and π: R → R/a the projection map. What is the image of π^*: Spec(R/a) → Spec(R)?

Q2. Show that Spec(R x S) = Spec(R) Spec(S) (where denotes the disjoint union).

Q3. For a p ∈ Spec(R), we define the residue field at p to be K(p) := Frac(R/p).

a) Prove that there exists a natural homomorphism α: R → K(p) whose kernel is ker α = p.

b) Pro β:  R → S is a homomorphism, then (β*)^-1 (p) ≅ Spec(S ⊗_R K(p)) (that is, find a bijection between the two sets).

Q4. Use the method outlined in Remark 2 to compute equation of independence of the following elements of C over Z:

√5 + sin(10^o) and (1+√5)/2 · 3√2.

Q5. Let d ≠ 1 be a squarefree integer. Show that the integral closure of Z in Q(√d) is

Q6. Let L/Q be a finite extension. Show that for any a ∈ L there exists 0 ≠ N ∈ Z such that Nα is integral over Z.

Q7. Prove Theorem 3.15 for general integral homomorphism is R → S.

Q8. Show that if R = K[X]/(f), where K is a field, and f is a polynomial with factorization f = ∏_iψ_i^e_i to irreducible polynomials, then Spec(R) = {x₁, . . . , x_n} = _i{x_i}, where {x_i} = Spec(K[X]/φ_i).

Q9. Let R = F_q[t₁, . . . , t_n] be the ring of polynomials of it variables and let v ≥ 1 be fixed.

a) Calculate #Hom(R, F_q^v).

b) Calculate the image of the map ψ: Hom(R, F_q^v) → Spec(R) given by ψ(α) = ker α.