Unformatted text preview: into nonunit elements. 2. Prove that Z [ i ] is a Euclidean domain (a ring with Euclidean algorithm) by exhibiting a function Î» from Z [ i ] to the nonnegative integers, and then mimicking the proof as done in class for R . 3. What are the units of Z [ i ]? (Prove your answer is correct). 4. Factor 2 into irreducible elements in Z [ i ] (or prove that it is, itself, irreducible â€“ i.e. not able to be nontrivially factored). 5. If Î± âˆˆ R , show that Î± must be congruent to one of 0,1, or 1 mod (1Ï‰ ). 6. Show that if, in R , N ( â„˜ ) 6 = 3 for a prime â„˜ , then 1 , Ï‰, and Ï‰ 2 are distinct in R/â„˜R . Conclude that 3  N ( â„˜ )1....
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 Spring '09
 BRUBAKER
 Algebra, Number Theory, Algebraic number theory

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