HINTS

Use the Euclid's algorithm for polynomials.

First prove that the leading coefficient of q is an integer. Then proceed by induction on the degree of q.

1. Use the definition to unravel both sides; note the convention about gcd's having leading coefficient 1.
2. Use the previous item and the extended Euclidean algorithm.

How could one divide X by 2X in Z/4Z?

If f(n)=p is a prime, consider f(n +kp).

First prove that an irreducible polynomial of degree > 1 cannot have zeros.

Use Fermat's Little Theorem

Write the polynomial p as p(x)= a_i x ²ⁱ+ b_i x ²ⁱ⁺¹. Then compare p(x) with p(-x).

Substitute r for x in a. Now look which terms are divisible by r and which terms are not. What can you conclude from that?

Remember that a reducible polynomial of degree 2 or 3 always has a zero.

Expand the left-hand side of the identity, and evaluate it in p/q.

Expand the product and argue, using all coefficients of the product, that the top coefficient of f is an integer.

• Split off a square: (X + 1)² + 1 and rewrite 1 as -2².
• Rewrite as X(X + 1) + 1 and show that substituting an integer mod 24 always leads to an odd number mod 24.
• Be careful: don't draw the conclusion that a zero should be a zero of at least one of the factors. Instead of trying each of the numbers 0, 1, ..., 11 mod 12 consider the question: when is the product of three consecutive integers divisible by 12?
• Replace the coefficients by even representatives (modulo 33) and get rid of the leading coefficient 2. Then split off a square. Be careful since 33 is not a prime.

If h(X) divides f(X), then h(X + a) divides f(X + a).

Find the gcd and write it as a combination of X - 1 and X² + X² + 1. Then divide both sides by X³ - 1.

Think of the proof of the theorem stating that there are infinitely many primes.

Move one term from the left-hand side to the right-hand side, then rewrite the new right-hand side as a product of two factors and compare with the factors on the left-hand side.