Comprehensive Exam: Math 780 & 784
Instructions: There are nine problems below (turn the sheet over for two of the problems).
Answer as many as you can on the blank pages provided with this test. You may keep the
questions when you are through.
1. Let m1,m2,b1, and b2be positive integers. Set d= gcd(m1, m2). Prove that the system
of congruences
x≡b1(mod m1)
x≡b2(mod m2)
has an integer solution if and only if b1≡b2(mod d).
2. Let a > 1 be an integer and let pbe an odd prime number.
(i) Let qbe a prime number such that qdivides ap−1. Prove that the order of amodulo
qis either 1 or p.
(ii) Explain why either q|(a−1) or q≡1 (mod 2p).
(iii) Prove that there are infinitely many primes which are 1 modulo 2p.
3. Let pbe a prime, and let hand kbe nonnegative integers such that h+k=p−1. Prove
that
h!k!+(−1)h≡0 (mod p).
4. Let n > 1 be an integer such that p= 2n+ 1 is a prime number.
(i) Prove that 3 is a primitive root modulo p.
(ii) Let abe a quadratic non-residue modulo p. Prove ais a primitive root modulo p.
5. Let aand bbe positive integers. Set d= gcd(a, b). Prove
gcd(2a−1,2b−1) = 2d−1.
6. Let Rbe the ring of algebraic integers in Q(√−7). Explain why Ris Euclidean.
7. For the following, f(x) = x3−10 and αis a root (any root) of f(x).
(a) Find a polynomial g(x) that has 3/(α−1) as a root.
(b) Is {1, α, α2}an integral basis for the ring of algebraic integers in Q(α)? Justify your
answer. (Hint: You should be able to see quickly that ∆(1, α, α2) is not squarefree.
This means that its value cannot be used in the obvious way to answer this question,
regardless of the answer. I suggest instead thinking about what part (a) has to do
with the question.)
