4 1. Let F be a finite field. Prove that there are infinitely many irreducible polynomials in F [x]. 2. Let p be a prime number. Prove that f (x) = xp1 + xp2 + . . . + x + 1 is irreducible in Q[x]. 3. Use the natural homomorphism from Z to Z5 to prove that x4 + 10x3 + 7 is irreducible in Q[x]. 4. Let n 1 and let Vn denote the set of complex numbers Vn = {z C | z n = 1 and z k 6= 1 0 < k < n}. a. What is the representation of the numbers z Vn using z = r(cos() + i sin())? (i.e. what are the values of r and for z Vn ?) b. Define the nth cyclotomic polynomial n (x) to be n (x) = Y (x z). zVn Prove that xn 1 = Y d (x) d|n c. Calculate 1 (x), 2 (x), 3 (x), 4 (x), 5 (x), 6 (x). d. Use the result from b. to prove that n (x) Z[x] for all n 1 (a-priori, n (x) is defined as a polynomial in C[x]). 1