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Unformatted text preview: 4400/6400 PROBLEM SET 0 0) Proof of Fermat’s Last Theorem, Step 0 : For a positive integer n , let FLT ( n ) denote the following statement: for x,y,z ∈ Z such that x n + y n = z n , xyz = 0. Show that if FLT(4) holds and FLT( p ) holds for each odd prime p , then FLT( n ) holds for all n ≥ 3. 1) Divisibility in Commutative Rings : For elements x,y of a commutative ring R , we define x | y (read as “ x divides y ”) to mean: there exists z ∈ R such that zx = y . This is binary relation on R . a) Give a complete description of the divisibility relation on the field of rational numbers Q . (Note: the point is that this is trivial.) For the remainder of this problem, let a,b,c be integers. b) Show that any integer a divides 0, but 0 divides only itself. c) Suppose a | b and a | c . Show that a | ( b + c ) and a | ( b- c ). d) Suppose a | b and a does not divide c . Show that a does not divide b + c . e) Suppose a does not divide b and a does not divide c . What can we conclude about whether a divides b + c ? 2) Divisibility Tests a) Show that 7 divides a positive integer 10 a + b if and only if 7 divides a- 2 b . Explain why this gives a test for divisibility by 7. b) Can you find a similar divisibility test for, say, 13? 3) Divisbility as a partial ordering : A relation R on a set is a partial or- dering if it satisfies the following axioms: (PO1) xRx for all x (reflexivity) (PO2) If xRy and yRx then x = y (anti-symmetry) (PO3) If xRy and yRz then xRz (transitivity) A partial ordering is total or linear if for any pair of elements x,y , either xRy or yRx holds (“ comparability ”). a) Note that the usual ≤ relation on the real numbers is a total ordering, hence endows every subset of the real numbers with a total ordering. In particular, the natural numbers N are totally ordered under ≤ . 1 b) For each of the following subsets of R , determine whether the divisibility relation is a partial ordering and/or a total ordering: (i) The set Z + of positive integers. (ii) The set N of non-negative integers. (iii) The set Z of all integers. 1 I’m not sure that there is anything to show here, but write a sentence or two to indicate that you understand the statement. 1 2 4400/6400 PROBLEM SET 0 4) Irrationality of √ 3: Adapt the proof given in class of the irrationality of √ 2 to show the irrationality of √ 3. (You can either phrase the argument in terms of infinite descent, or start by reducing to lowest terms, according to your taste.) Make sure you give a complete proof of the fact that 3 | x 2 implies 3 | x and that your proof does not assume the uniqueness of factorization into primes. 4.5) Irrationality of √ p : It is in fact true that for any prime number p , √ p is irrational....
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This note was uploaded on 10/26/2011 for the course MATH 4400 taught by Professor Staff during the Spring '11 term at University of Georgia Athens.
- Spring '11
- Number Theory