h3 115A - norm of , denoted N ( ), as N ( ) = a 2 + 5 b 2 ....

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MAT 115A University of California Fall 2010 Homework 3 due October 20, 2010 1. Rosen 3.3 #20, pg. 95 Let a 1 ,a 2 ,...,a n be integers not all equal to zero. Is it true that the greatest common divisor of these integers ( a 1 ,...,a n ) is the least posi- tive integer of the form m 1 a 1 + m 2 a 2 + ··· + m n a n where m 1 ,...,m n Z ? If so, prove it. If not, give a counterexample. 2. Rosen 3.5 #8, pg. 117 Show that every positive integer can be written as the product of pos- sibly a square and a square-free integer. A square-free integer is an integer that is not divisible by any perfect squares other than 1. 3. Rosen 3.5 #19, 22, 23, pg. 117 Let α = a + b - 5 where a,b Z . Define the
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Unformatted text preview: norm of , denoted N ( ), as N ( ) = a 2 + 5 b 2 . (a) Show that if = a + b -5 and = c + d -5, where a,b,c,d Z , then N ( ) = N ( ) N ( ). (b) Show that the numbers 1 + -5 and 1--5 are prime numbers, that is, there are no numbers = a + b -5 and = c + d -5 dierent from 1 such that 1 -5 = . (Hint: Use part (a)). (c) Find two dierent factorizations of the number 21 into primes of the form a + b -5, where a and b are integers. 4. Rosen 3.5 #45, pg. 119 Show that 2 + 3 is irrational. 1...
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