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.
 Spring '10
 anne
 Number Theory, Integers

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