Attempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt additional questions.

Transcription

1 UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination FERMAT S LAST THEOREM MTHD6024B Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt additional questions. You will not be Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. MTHD6024B Module Contact: Professor Shaun Stevens, MTH Copyright of the University of East Anglia Version: 1

2 - 2 - Note: You may use any theorems you know from books or from lectures, provided you make it clear which result you are using, except those theorems you are required to prove. You may also use standard facts from group theory, ring theory and linear algebra. In this exam, 0 is not a natural number. 1. This question concerns the diophantine equation ( ) x 2 + 6y 2 = z 2. Let x, y, z be coprime natural numbers satisfying ( ). (i) Prove carefully that x, y, z are pairwise coprime. (ii) Prove that x, z are odd and y is even. (iii) Prove that there are natural numbers a, b such that x = a 2 6b 2, y = 2ab, z = a 2 + 6b 2, (a, 6b) = 1, or x = 3a 2 2b 2, y = 2ab, z = 3a 2 + 2b 2, (3a, 2b) = 1. Hence find the solution to ( ) with x, y, z natural numbers and minimal value of z. [10 marks] 2. (i) Let n be a natural number and let d be an integer. Define carefully what is meant by a reduced positive definite binary quadratic form of discriminant d and what it means to say that a binary quadratic form f properly represents n. (ii) Find all reduced binary quadratic forms of discriminant 36. [6 marks] [6 marks] (iii) Let p 5 be a prime number. Prove that there exist natural numbers x, y such that p = x 2 + 9y 2 if and only if p 1, 13, 25 (mod 36). [8 marks] [You may find it useful to observe that the squares modulo 9 are precisely 0, 1, 4, 7.] MTHD6024B Version: 1

3 This question concerns the diophantine equation ( ) x 4 + 6y 4 = z 4, with x, y, z natural numbers. (i) Suppose x, y, z are natural numbers satisfying x 4 + 6y 4 = z 2. (a) Suppose that x, y, z are not pairwise coprime. Prove that there are natural numbers u, v, w, with u 4 + 6v 4 = w 2, and w < z. (b) Suppose that x, y, z are pairwise coprime. Prove that there are natural numbers u, v, w, with u 4 + 6v 4 = w 2, and w < z. [10 marks] [You are given that the natural number solutions to the equation x 2 +6y 2 = z 2, with x, y, z coprime, are given by x = a 2 6b 2, y = 2ab, z = a 2 + 6b 2, (a, 6b) = 1, or x = 3a 2 2b 2, y = 2ab, z = 3a 2 + 2b 2, (3a, 2b) = 1, with a, b natural numbers. You may find it useful to consider congruences modulo 3 or 8 to rule out cases.] (ii) Explain briefly the method of descent and prove that the equation ( ) has no solutions. MTHD6024B PLEASE TURN OVER Version: 1

4 This question concerns the diophantine equation ( ) x y 2 = 2z 3. We set K = Q( 23). (i) Define the ring of integers o K of K. Give (without proof) a Z-basis of the form {1, α} for o K, and write down the minimum polynomial f α (X) Z[X] of α. (ii) Let p be a prime. Give a criterion (using Dedekind s Theorem or otherwise) by which one can deduce the factorization of the ideal (p) in o K. (iii) Prove that (2) factorizes in o K as (2) = p 2 p 2, where p 2 p 2 and both are nonprincipal. (iv) Given that the class number of K is 3, prove that there is no solution [x, y, z] N 3 to ( ). MTHD6024B Version: 1

5 Let p be an odd prime number, let ζ be a fixed primitive p th root of unity and put π = 1 ζ. Put K = Q(ζ) and denote by o K the ring of integers of K. You may assume that o K = Z[ζ] = Z[π] and that K has degree p 1 over Q. (i) Let l be a prime number. State a criterion by which one can compute the factorization of (l) in o K, in terms of l (mod p). [You should consider the cases l = p and l p separately.] (ii) Suppose α o K. Prove that, if there exists β o K such that α = β p then there exists a Z such that α a (mod p), and explain briefly whether the converse of this statement is true. Now set p = 11. (iii) Compute the factorizations of (3) and (13) in o K. (iv) Set F = Q( 11) and write o F for its ring of integers. Prove that F K and find a prime number l which is inert in o F but not inert in o K. MTHD6024B PLEASE TURN OVER Version: 1

6 Let p, q be distinct odd prime numbers. Let ζ = e 2πi/p be a primitive p th root of unity and put F = Q(ζ), with ring of integers o F = Z[ζ]. (i) What does it mean to say that p is regular? What does it mean to say that q is inert in o F? Now set p = 11 and q = 7 ; you may assume that 11 is a regular prime and 7 is inert in o F. Consider the equation ( ) x 11 + y 11 = 7z 11, x, y, z Z pairwise coprime, 11 xyz. (ii) Prove that (x+ζ i y), (x+ζ j y) are coprime ideals, for 0 i < j 10. (iii) Explain why (7) divides some factor (x + ζ i y). By considering complex conjugation, or otherwise, prove that (7) (x + y). (iv) Deduce that there are a unit ε and α o F such that x + ζy = εα 11. By taking complex conjugates and looking modulo 11, prove that x y (mod 11) and deduce that there are no solutions to ( ). [You may use the fact that ε/ε = ζ r, for some r. You may also use that the 11 th powers modulo 121 are 0, 1, 3, 9, 27, 40, 81, 94, 112, 118, 120.] END OF PAPER MTHD6024B Version: 1