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Unformatted text preview: Math 5125 Monday, October 31 Ninth Homework Solutions 1. Let R = Z [  n ] where n is a square free integer greater than 3. (a) Prove that 2,  n and 1 +  n are irreducibles in R . (b) Prove that R is not a UFD. (c) Give an explicit ideal in R that is not principal. Define N : R Z by N ( a + b  n ) = ( a + b  n )( a b  n ) = a 2 + nb 2 . Note that N ( ) = N ( ) N ( ) for , R . Also if N ( ) = 1, then we must have N ( ) = 1, a = 1 and b = 0, which tells us that = 1 and in particular is a unit. (a) N ( 2 ) = 4, so if 2 = with , nonunits, then N ( ) = 2. This is not possible because we cannot solve a 2 + nb 2 = 2 with a , b Z . Furthermore N (  n ) = n so if  n = , then N ( ) N ( ) = n which implies either N ( ) or N ( ) = 1 and we deduce that either or is a unit. Finally N ( 1 +  n ) = 1 + n , so if 1 +  n = , then again we must have either N ( ) or N ( ) = 1, which shows that either or is a unit. We have now proven that 2,  n and 1 +...
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 Fall '07
 PALinnell
 Math, Algebra

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