ahw9 - Math 5125 Monday, October 31 Ninth Homework...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the 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

Page1 / 2

ahw9 - Math 5125 Monday, October 31 Ninth Homework...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online