This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
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 +...
View Full Document