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Unformatted text preview: ms 2 . (a) Show that N ( xy ) = N ( x ) N ( y ). (b) Show that N ( x ) Z if x R m . (c) Show that u U ( R m ) if and only if N ( u ) = 1. (d) Use (c) to show that U ( R1 ) = { 1 , i } , U ( R3 ) = { 1 , (1 3) / 2 } , and U ( R m ) = { 1 } for all other negative squarefree m in Z . (4) Let a and b be nonzero elements of a Euclidean domain such that a  b and d ( a ) = d ( b ). Show that a and b are associates. (5) Prove that if m =3,7, or11, then R m is Euclidean with d ( r ) =  N ( r )  for all nonzero r R m . [Hint: Mimic the proof of Proposition 3.7 from class, but choose d Z nearest to 2 t and then c Z so that c is as near to 2 s as possible with c d (mod ) , then set q = ( c + d m ) / 2.]...
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 Fall '08
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 Algebra

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