Unformatted text preview: ( 1 2 ) 2 = 1 4 / ∈ R . (iii) Using the fact that α = 1 2 ( 1 + √ − 19 ) is a root of x 2 − x + 5, prove that R = { a + b α : a , b ∈ Z } is a domain. Solution. It is clear that R contains 1, and one shows easily that if a , a ∈ A and b , b ∈ B , then R contains a − a , b − b , a − b , and aa . Write a = α + α √ 5, b = 1 2 (β + β ) √ 5, and b = 1 2 (γ + γ ) √ 5. It is easy to see that ab ∈ A if α and α have the same parity, while ab ∈ B otherwise; in either case, ab ∈ R . Finally, write bb = h 1 2 (β + β ) √ 5 ih 1 2 (γ + γ ) √ 5 i = 1 4 h (βγ + 5 β γ ) + √ 5 (βγ + β γ) i . Expand and substitute β = 2 p + 1, β = 2 p + 1, γ = 2 q + 1, and γ = 2 q + 1 (for β , β , γ , and γ are odd). After collecting terms, one sees that both the constant term and the coef f cient of...
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
 Fall '11
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