Exercise 1

Determine the gcd of each of the following pairs of polynomials and write the gcd as a combination of the given polynomials.

1. X² + 1 and X³ + 1 in Q[X].
2. X² + 1 and X³ + 1 in Z/2Z[X].
3. X² - X + 1 and X³ + X + 2 in Z/3Z[X].

Exercise 2

Suppose that the polynomials a, b are from Z[X] and that b is monic, i.e., has leading coefficient 1. Prove that the quotient q and remainder r of division of a by b in Q[X] also belong to Z[X].

Exercise 3

Analogous to the definition of the gcd of two polynomials one can define the gcd of more than two (nonzero) polynomials. Let a, b, c be three non-zero polynomials in Q[X].

1. Show that gcd(gcd(a,b),c) = gcd(a,b,c).
2. Show that a, b, c are relatively prime (have gcd 1) if and only if there exist polynomials p, q, r such that
   pa + qb + rc = 1.

Exercise 4

Explain why division with remainder fails in Z/nZ[X] when n is an integer but not a prime.

Exercise 5

Let f Z[X] be a polynomial of degree 1. Prove that f(n) cannot be a prime for each n Z.

Exercise 6

Which of the following polynomials is irreducible and why?

• X³ + 8 in Q[X],
• X³ + X + 1 in Z/2Z[X],
• X⁴ + X + 1 in Z/2Z[X],
• X⁴ + 2X² + 1 in C[X],
• (X² + X + 1)³ + (X⁷ - 1)³ in Q[X],
• (X² - 1) ⁹⁷ + (X³ - 1)⁷⁸ in Z/5Z[X].

Exercise 7

Show that for any prime p and any polynomial a₀ + a₁X + ··· + a_mX^m Z/pZ[X] we have

(a₀ + a₁X + ··· + a_mX^m)^p = a₀^p + a₁^pX^p + ··· + a_m^pX^mp.

Exercise 8

Find all polynomials p Q[X] that satisfy p(x) = p(-x) for any x. Answer the same question if we replace Q by Z/6Z or Z/2Z.

Exercise 9

Consider the polynomial a = a₀ + ··· + a_{n - 1}X^{n - 1} + a_nXⁿ Z[X], with a_n 0.

1. Prove: If r Z is a zero of a then r is a divisor of a₀.
2. Suppose that r, s Z are relatively prime and that r/s is a root in Q of a. Prove that s divides a_n and that r divides a₀.
3. Find all rational roots of the polynomial 15 - 32X + 3X² + 2X³.

Exercise 10

1. How many polynomials of degree n are there in Z/3Z[X]?
2. Determine all irreducible polynomials in Z/3Z[X] of degrees 2 and 3.

Exercise 11

Verify the identity of polynomials

(X² - 1)² + (2X)² = (X² + 1)².

A Pythagorean triple is a triple of positive integers r, s and t such that r² + s² = t². According to the Pythagorean theorem, these triples occur as sides of right triangles. By substituting rational numbers p/q for X show how to produce Pythagorean triples from the identity (X² - 1)² + (2X)² = (X² + 1)².

Exercise 12

Let f be a polynomial in Q[X] such that the product (4X - 3) · f is an element of Z[X]. Prove that all coeffients of f are integers.

Exercise 13

Find all zeros of each of the following polynomials

• X² + 2X + 2 in Z/5Z.
• X² + X + 1 in Z/24Z.
• X(X + 1)(X + 2) in Z/12Z.
• 2X² + 13X + 9 in Z/33Z.

Exercise 14

Suppose the polynomials f(X) and g(X) in Q[X] have gcd d(X). Fix a in Q and replace every occurrence of X in f and g by X + a. For instance, if a = 2 then X² + X - 1 changes into (X + 2)² + (X + 2) - 1. Prove that the gcd of the new polynomials f(X + a) and g(X + a) is d(X + a).

Exercise 15

Show that the polynomials X - 1 and X² + X² + 1 in Q[X] have gcd 1. Use the extended Euclidean algorithm to find constants a, b and c such that

3/(X³ - 1) = a/(X - 1) + (bX + c)/(X² + X + 1).

Exercise 16

Let R be one of the fields Q, R, C or Z/pZ with p prime. Prove that there are infinitely many irreducible polynomials in R[X].

Exercise 17

Determine all irreducible polynomials p and q in Q[X] with integer coefficients that satisfy the equation

(X² + 1)p + (X + 2)q = pq.