Adv Alegbra HW Solutions 86

Adv Alegbra HW Solutions 86 - 1 2 2 = 1 4 ∈ R(iii Using...

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86 (iii) Prove that the ring of Gaussian integers is a domain. Solution. The set Gaussian integers form a subring of C , and hence it is a domain. 3.12 Prove that the intersection of any family of subrings of a commutative ring R is a subring of R . Solution. Absent. 3.13 Prove that the only subring of Z is Z itself. Solution. Every subring R of Z contains 1, hence 1 + 1 , 1 + 1 + 1, etc, so that R contains all positive integers (one needs induction), and f nally, R contains the additive inverses of these, i.e., all negative integers, as well. 3.14 Let a and m be relatively prime integers. Prove that if sa + tm = 1 = s 0 a + t 0 m , then s s 0 mod m . Solution. The given equation gives the congruences as 1mod m and as 0 1mod m .Now a and m relatively prime, and Theorem 1.69 says that any two solutions to ax 1mod m are congruent; that is, s s 0 mod m . 3.15 (i) Is R ={ a + b 2 : a , b Z } a domain? Solution. It suf f ces to show that R contains 1 and is closed under addition and multiplication. Each of these is routine: for example, ( a + b 2 )( c + d 2 ) = ( ac + 2 bd ) + ( ad + bc ) 2 . (ii) Is R ={ 1 2 ( a + b 2 ) : a , b Z } a domain? Solution. R is not a subring of R , hence is not a domain, for
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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.

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