# Algebraic Number Theory Help

• Nov 18th 2007, 09:04 PM
Susan
Algebraic Number Theory Help
I am stuck on the following.

Let R={(a+b(-163)^1/2)/2 : a,b are integers and a-b is even}. We know that R is a PID.

Let z=(a+b(-163)^1/2)/2 then show that the cardinality of R/(z) is

(a^2+b^2(163))/4

this is just the magnitude of z squared but I am not sure how to arrive at the conclusion. Any help would be appreciated.

Susan
• Nov 20th 2007, 07:16 PM
ThePerfectHacker
Quote:

Originally Posted by Susan
I am stuck on the following.

Let R={(a+b(-163)^1/2)/2 : a,b are integers and a-b is even}. We know that R is a PID.

Let z=(a+b(-163)^1/2)/2 then show that the cardinality of R/(z) is

It makes no sense to let $\displaystyle z=\frac{a}{2}+i\frac{b}{2}\sqrt{163}$, how is that well defined? Note $\displaystyle z$ has to be a specifc number in $\displaystyle R$ because since it is a PID any ideal must have the form $\displaystyle \left< z \right>$ where $\displaystyle z$ is some particular element in $\displaystyle R$.