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END OF SEMESTER EXAM, MP473, 1992
Time: Three hours Candidates should aim to complete SIX questions, but may attempt as many questions as they wish.
(In what follows, A(
d) denotes the ring of integers in Q(
d).)
- (a) Define the term algebraic integer and prove that a complex number θ is an algebraic integer if and only if ∃ w 1 ,... , wn, not all zero, such that for 1 ≤ i ≤ n,
wiθ =
∑^ n
j=
aij wj ,
where aij ∈ Z for 1 ≤ i ≤ n, 1 ≤ j ≤ n.
(b) If θ is an algebraic integer and is also a rational number, prove that θ is an integer.
(c) Prove that the sum and product of two algebraic integers is also an algebraic integer.
- (a) Prove that the ring Z[i] of Gaussian integers is Euclidean.
(b) Determine the units of Z[i].
(c) Describe the factorization of a prime p into irreducibles of Z[i].
(d) Determine the factorization of 6 + 7i into irreducibles of Z[i].
- (a) If k is an odd rational integer, prove that
gcd (k + i, k − i) = 1 + i.
(b) Show that the only solutions of the Diophantine equation
x 2
are x = ± 1 , y = 1.
- An integer α > 0 of Q(
d), d > 0 , is called primary if
α
σ(α)
∣ < η
2 ,
where η is the fundamental unit of Q(
d).
(a) Prove that every non–zero integer of Q(
d) is the associate of precisely one primary integer.
(b) Prove that the primary integers α with N (α) = n satisfy
α 2 − Aα + n = 0,
where |A| <
|n|(1 + η).
(c) Find the primary integers of Q(
- with norm equal to 7 and hence find all solutions in integers of x^2 − 2 y^2 = 7.
- (a) If p is a prime of the form 3n + 1, use the fact that the integers of
Q(
−3) form a UFD to prove that p = x^2 − xy + y^2 is soluble in integers x and y. How many solutions are there?
(b) If p is a prime of the form 8n ± 1, use the fact that the integers of Q(
- form a UFD to prove that p = x^2 − 2 y^2 is soluble in integers x and y. (Hint: η = 1 +
2 is the fundamental unit and N (η) = −1.
- (a) Prove Hurwitz’ lemma: Let α, β ∈ A(
d), g|N (α), g|N (β), g|(ασ(β)+ βσ(α)), where σ(α) is the conjugate of α. Prove that g|ασ(β). (HINT: ξ = ασ(β) satisfies the equation
ξ 2 − T (ξ)ξ + N (ξ) = 0.)
(b) Use Hurwitz’ lemma to prove that if A is an ideal of A(
d), then
Aσ(A) = (g),
where g ∈ N.
(c) Also prove that if A and C are ideals in A(
d), then
A|C ⇔ A ⊇ C.
- Let p be a prime, d a squarefree integer, ω = (1 +
d)/2 if d ≡ 1 (mod 4), but
d otherwise. Also let f be the defining polynomial of ω. Let A = (p, a + ω), where a ∈ Z.
(a) Prove that A = (1) if f (−a) 6 ≡ 0 (mod p).
(HINT: gcd (x + a, f ) = 1 in Zp[x].)
- (a) Find the group structure of the multiplicative group of equivalence classes of ideals in A(
(b) Let d > 0 and squarefree, (α) = A 2 , where A is an ideal in A(
d), N (α) < 0 and N (η) = 1, where η is the fundamental unit. Prove that A is not principal.
(c) Consider the ideal A = (3, 1 +
34). Prove that A^2 = (−5 +
and hence prove that A is not principal, given that 35 + 6
34 is the fundamental unit of Q(
- Let m > 0 and squarefree.
(a) Prove that A(
−m) is not a UFD if one of the following hold:
(i) m ≡ 1 (mod 4), m > 1;
(ii) m ≡ 2 (mod 4), m > 2;
(iii) m ≡ 7 (mod 8), m > 7.
(b) If A(
−m) is a UFD and m ≡ 3 (mod 8), prove that m is a prime and that x^2 + x + m+ 4 assumes prime values for^ x^ = 0 , 1 ,... , m−^3 4
. (These are Euler’s prime–producing polynomials.)
(c) Suppose that m ≡ 3 (mod 8), m is a prime and that x 2 +x+ m+ 4 assumes prime values for x = 0, 1 ,... , m 4 − 3. Prove that A(
−m) is a UFD by showing that all ideals are principal.
- Do one of the following only:
(a) Use the Gaussian sum identity G(∆) =
∆ to explicitly evaluate the series ∑∞
n=
n
n
(b) Sketch a proof of the formula G(p∗) =
p∗, where p is an odd
prime and p ∗ = (−1)
p− 1 (^2) p.
