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# sol9 - Algebra Assignment 9 Solutions 1 Let R be a ring...

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Unformatted text preview: Algebra , Assignment 9 Solutions 1. Let R be a ring. Call a ∈ R a unit if ab = 1 for some b ∈ R . Let X be the set of units Prove that X forms a group under multiplication. What is our standard notation for X in the case where R = Z n ? Solution: Identity: As 1 · 1 = 1, 1 ∈ X . Product: Let a, c ∈ X . Then there exist b, d ∈ R with ab = cd = 1. Then ( ac )( bd ) = ( ab )( cd ) = 1 · 1 = 1 so ac ∈ X . Inverse: Let a ∈ X . Then there exists b ∈ R with ab = 1. But then b ∈ X as there exists an element (namely, a ) whose product with b is 1. 2. Recall Z [ √ 2] = { a + b √ 2 : a, b ∈ Z } . Find a unit (as defined above) a + b √ 2 ∈ Z [ √ 2] which has a ≥ 10. (One approach: Find some unit other than ± 1 and apply previous problem.) Solution: By a combination of luck, skill and diligence you notice that ( √ 2 + 1)( √ 2 − 1) = 1 so that α = 1 + √ 2 is a unit. But then any power of α is a unit. α 2 = 3+2 √ 2, α 3 = 7+5 √ 2 and α 4 = 17+12 √ 2 is as desired. (There are other solutions as well.) 3. Let Z [ i ], as usual, denote the Gaussian Integers and set I equal the multiples of 2 + i . (That is, I is all numbers (2 + i ) α where α ∈ Z [...
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sol9 - Algebra Assignment 9 Solutions 1 Let R be a ring...

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