Discrete Algebraic Methods

8.16. Let A ∈ PSL(2, ℤ) be an element in the modular group of order 2 and Furthermore, let S ∈ PSL(2, ℤ) with

(a)Show that a + d = 0.

(b)Show that A and S are conjugated in PSL(2, ℤ).

Hint: Use the normal form Ri1S ⋅⋅ ⋅ Rim−1SRim , m ≥ 2, 1 ≤ ij ≤ 2 for j = 1, . . . , m and i1 = im, as in the proof of Theorem 8.38.

8.17. Let n ∈ ℕ, n ≥ 1. Prove Fermat’s theorem on sums of two squares:

(a)If −1 is a quadratic residue modulo n, that is, −1 ≡ q2 mod n for some q ∈ ℤ, then n is a sum of two squares in ℤ, that is, n = x2 + y2 with x, y ∈ ℤ.

Hint ...