Problem 2. Let p be a prime number. (a) (5 pts) Let f(T) be a polynomial...

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Problem 2. Let p be a prime number. (a) (5 pts) Let f(T) be a polynomial modulo p of degree 2 or 3. Prove that f(T) is irreducible if and only if f(T) has no roots modulo p. Hint. Prove the contrapositive, looking at the degrees of the divisors of f(T). (b) (5 pts) Count the number of monic polynomials modulo p of degree d. (c) (5 pts) Count the number of monic irreducible polynomials modulo p of degree 2. (d) (5 pts) Count the number of monic irreducible polynomials modulo p of degree 3. Problem 2. Let p be a prime number. (a) (5 pts) Let f(T) be a polynomial modulo p of degree 2 or 3. Prove that f(T) is irreducible if and only if f(T) has no roots modulo p. Hint. Prove the contrapositive, looking at the degrees of the divisors of f(T). (b) (5 pts) Count the number of monic polynomials modulo p of degree d. (c) (5 pts) Count the number of monic irreducible polynomials modulo p of degree 2. (d) (5 pts) Count the number of monic irreducible polynomials modulo p of degree 3.