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

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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 α , denoted N ( α ), as N ( α ) = a 2 + 5 b 2 . (a) Show that if α = a + b √-5 and β = c + d √-5, where a,b,c,d ∈ Z , then N ( αβ ) = N ( α ) N ( β ). (b) Show that the numbers 1 + √-5 and 1-√-5 are prime numbers, that is, there are no numbers α = a + b √-5 and β = c + d √-5 diﬀerent from ± 1 such that 1 ± √-5 = αβ . (Hint: Use part (a)). (c) Find two diﬀerent 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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This note was uploaded on 10/23/2010 for the course MATH math 115A taught by Professor Anne during the Spring '10 term at UC Davis.

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