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NT2009HW4

# NT2009HW4 - 4400/6400 PROBLEM SET 3 A sucient number of...

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4400/6400 PROBLEM SET 3 A sufficient number of problems: 4 for 4400 students, 6 for 6400 students. 3.1) Let c and N > 1 be integers, and let c be the class of c modulo N . a) Show that c is a unit in Z /N Z if and only if gcd( c, N ) = 1. b) Show that #( Z /N Z ) × N - 1, with equality holding if and only if N is prime. 3.2) Let m and b be real numbers, and consider the line : y = mx + b. a) Show that the only possibilities for the number of Q -rational points ( x, y ) on are: none, exactly one, infinitely many. b) Suppose m and b are both rational. Show has infinitely many rational points. c) Suppose m is rational and b is irrational. Show has no rational points. d) Suppose m is irrational and b is rational. Show has exactly one rational point. e) What can be said when m and b are both irrational? 3.3) (Converse of Wilson’s Theorem) Let N > 1 be such that ( N - 1)! ≡ - 1 (mod N ) . Show that N is prime. 3.4)* Let D be a squarefree integer which is not 0 or 1, and put R D := Z [ D ] = { a + b D | a, b Z } = Z [ t ] / ( t 2 - D ) . Thus R D is the subring of the complex numbers obtained by adjoining D to Z . a) Show that R D is an integral domain, with fraction field Q [ D ] = { a + b D | a, b Q } .

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NT2009HW4 - 4400/6400 PROBLEM SET 3 A sucient number of...

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