ASSIGNMENT 5 – ELEMENTARY NUMBER THEORY MATH 392

Eyal Goren, November 11, 1998

1. Let ϕ : G_{1} ⟶ G_{2} be a homomorphism of groups.

(a) Prove that if H is a subgroup of G_{1} then ϕ(H) is a subgroup of G_{2}.

(a) Prove that if H is a subgroup of G_{2} then ϕ^{−1}(H) is a subgroup of G_{1}, where ϕ^{−1}(H) = {g ∈ G_{1} : ϕ(g) ∈ H}.

2. Let G be a group. Prove that

∑_{g∈G} g = ∑_{g∈G; ord(g)=2} g.

Conclude Wilson’s theorem: Let p be a prime number then

(p − 1)! ≡ −1 (mod p).

3. Let R be a ring.

(A) Given elements a_{1}, ⋯, a_{r} of R. Consider the set

Ra_{1} + Ra_{2} + ⋯ + Ra_{r}

and denote it by (a_{1}, ⋯, a_{r}). Prove it’s an ideal.

(B) Let I and J be two ideals of R and let IJ denote the set of all finite sums d_{1} + ⋯ + d_{t} in which each d_{ℓ} is a product of an element from I and an element from J. That is,

IJ = {d_{1} + ⋯ + d_{t} : ∀ℓ, ∃(i_{ℓ} ∈ I, j_{ℓ} ∈ J), d_{ℓ} = i_{ℓ}j_{ℓ}} .

Prove that IJ is an ideal of R.

(C) Prove that if I = (a_{1}, ⋯, a_{α}) and J = (b_{1}, ⋯, b_{β}), then IJ = (c_{11}, ⋯, c_{ij}, ⋯, c_{αβ}), where c_{ij} = a_{i}b_{j}.

4 Write the proof of Lagrange’s theorem on sum of squares in complete detail.

5 (A) Let p be a prime and f(x) be a polynomial of degree d and coefficients in ℤ/pℤ. Prove that f has atmost d roots.

(B) Show by an example that if p is not a prime, the assertion needs no longer be true.

6 Write a fully detailed proof of the Quadratic Reciprocity Law, following Baker’s book. In particular, check that the transformation

x = \frac{1}{2}(q + 1) − x′; y = \frac{1}{2}(p + 1) − y′,

appearing there, does give a bijection between the integer points in the regions appearing in the proof.

7 Find the number of solutions for the following equations in ℤ/65537ℤ (65537 is a prime number...):

(A) x^{2} + y^{2} = −1.

(B) x^{2} + y^{2} = 3.

(C) 2x^{2} + y^{2} = 1.