Comprehensive Exam, Summer 2003
for
MATH 780 & MATH 784
1. Let p≡1 (mod 4) be a prime number. Define Rto be the set of integers in {1,2, . . . , p −1}which are quadratic
residues modulo p, and Nto be the set of integers in {1,2, . . . , p −1}which are quadratic nonresidues modulo p.
(For example, if p= 5, then R={1,4}, and N={2,3}.) Prove that
X
n∈R
n=X
n∈N
n.
(In other words, show that the sum of the elements of the set Requals the sum of the elements of the set N.)
2. Let a≥2and b≥2be relatively prime integers. Let Sbe the set of positive integers which can be represented
in the form ak +bl with kand lnonnegative integers. (For example, it is not difficult to show that if a= 2 and
b= 3, then S={2,3,4,5, . . .}.) Prove that the largest positive integer which is not in Sis ab −a−b. (Hint: If
nis an integer, then at least one of the numbers n, n −a, . . . , n −(b−1)ais divisible by b.)
3. Let nbe a positive integer.
(a) Find the order of 2modulo 22n+ 1.
(b) Let pbe a prime divisor of 22n+ 1. Prove that p≡1 (mod 2n+1).
(c) Let kbe a positive integer. Using (b), prove that there exist infinitely many primes which are congruent to 1
modulo 2k+1.
4. Let pbe a prime number.
(a) Suppose that pis odd and is not of the form 8k+ 5 with kan integer. Prove that the congruence a4≡4
(mod p)has an integer solution.
(b) Suppose that pis of the form 8k+ 5 with kan integer. Prove that the congruence a4≡ −4 (mod p)has an
integer solution.
(c) Prove that for any prime pthe congruence a8≡16 (mod p)has an integer solution.
5. Let (xn, yn)denote the nth positive integral pair (x, y)satisfying x2−2y2= 1 (ordered so that x1< x2<·· ·).
Thus, for example, (x1, y1) = (3,2) and (x2, y2) = (17,12). Let pbe a prime. Prove that
xp≡x1(mod p)and yp≡y1(mod p)⇐⇒ p≡ ±1 (mod 8).
(Comment: This is meant to test your knowledge of both Math 780 and Math 784.)
6. Prove the following theorem from class, giving as many details as you can.
Theorem Let αbe an algebraic number with minimal polynomial f(x) = xn+Pn−1
j=0 ajxj∈Q[x]. Every
element of Q(α)can be expressed uniquely in the form g(α)where g(x)∈Q[x]with g(x)≡0or
deg g(x)≤n−1.
