# Rings: Principal Ideal domain

• Mar 7th 2010, 10:23 AM
sarah89
Rings: Principal Ideal domain
Hey Guys,

I've never been on one of these sites before but am really stuck on an problem sheet I have been set. I'm doing fine in all my other modules but this algebra one doesn't seem to click well with me :(
Any help would be much appreciated.

So I have to prove that R is a principal ideal
domain where R is the ring {a + b(sqrt(-2)) | a,b are integers}

(by b(sqrt(-2)) I mean b multiplied by the square root of -2)

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• Mar 7th 2010, 06:21 PM
NonCommAlg
Quote:

Originally Posted by sarah89
Hey Guys,

I've never been on one of these sites before but am really stuck on an problem sheet I have been set. I'm doing fine in all my other modules but this algebra one doesn't seem to click well with me :(
Any help would be much appreciated.

So I have to prove that R is a principal ideal
domain where R is the ring {a + b(sqrt(-2)) | a,b are integers}

(by b(sqrt(-2)) I mean b multiplied by the square root of -2)

define the map $f: R \setminus \{0 \} \longrightarrow \mathbb{N}$ by $f(a+b \sqrt{-2})=a^2+2b^2.$ show that $f$ is a norm-Euclidean and so $R$ is an Euclidean domain and therefore a PID.