Pries: 676 Number Theory. 2010. Homework 1.
Let p be an odd prime.
Introduction
1. (IR 5.14) If x3 = 1 and x 6= 1, show that (2x + 1)2 = −3. Give a direct proof that
x2 ≡ −3 mod p has a solution when p ≡ 1 mod 3 using that (Z/p)∗ is cyclic.
(Next week we will show that x2 ≡ −3 mod p has a solution if and only if p ≡ 1 mod 3,
which may be useful in the following problems).
2. (a) Show there are infinitely many primes congruent to 2 modulo 3.
(b) Show there are infinitely many primes congruent to 1 modulo 3.
√
3. Show that Z[ −3] does not have unique factorization but that O−3 = Z[ζ3 ] does.
4. Show that p factors in O−3 if and only if p ≡ 1 mod 3.
Finite fields
1. Prove Wilson’s Theorem (p − 1)! ≡ −1 mod p.
2. Suppose α is not a square in a finite field K. Let L be a field extension of K with
degree d. Show that α is a square in L if and only if d is even.
3. (IR 7.15) Consider h(x) = xn − 1 ∈ Fq [x] with gcd(n, q) = 1. Show h(x) splits into
linear factors in some extension of Fq and the least degree of such an extension is the
smallest positive integer r so that q r ≡ 1 mod n.
4. (DF 14.12) Let q = pn . The Frobenius morphism σp is an Fp -linear transformation of
Fq . Find its characteristic polynomial. Show that σp is diagonalizable over Fp if and
only if p ≡ 1 mod n and is diagonalizable over some extension of Fp if and only if p - n.