Prove that if p is a prime in Z that can be written in the form a2 + b2,then a + bi is irreducible in Z[i]. Find three primes that have thisproperty and the corresponding irreducibles.

Suppose that m is a composite natural number and let n > 0 be any integer. Find apolynomial of degree n in Zm[x] that is a zero divisor.

Show that if p and q are primes, then the cyclotomic polynomial Øq(x) = 1 + x + ... + x9−¹ has q – 1 roots in Z₂ if p = 1 mod (q), has one if p = q, and has none otherwise. р

(3) Let p be a prime. Show that Z = Zp\ {0}.

Show that x4 – 22 x2 +1 is irreducible over ℚ. [Hint: Show that it has no factor of degree 1 and degree 2.]

Let m and n be positive integers such that (m, n) = 1. Prove that if m and n are both perfect, then mn is 4-perfect.

Prove that if p > 2 is a prime, then (1 2) and (1 2... p-1 p) generate Sp.

7. Let a be a root of X² - bX+c, for b, c integers such that b² - 4c < 0. Suppose that unique factorisation holds in Z[a]. Let m, n be relatively prime positive integers such that mn is expressible as x² + bxy + cy² for integers x and y. Show that both m and n are expressible in this form.