1. ## Principal Ideal

How do I show the ideal $\displaystyle$I = \left( {1 - \sqrt { - 5} } \right)$$is principal in \displaystyle {Z}\left[ {\sqrt { - 5} } \right$$

Thanks!

2. Originally Posted by orbit
How do I show the ideal $\displaystyle$I = \left( {1 - \sqrt { - 5} } \right)$$is principal in \displaystyle {Z}\left[ {\sqrt { - 5} } \right$$

Thanks!

You should really try to think over your questions before you send them: in ANY ring, the ideal <x> is principal by DEFINITION (read it)

Tonio

3. Yeah.... are you sure you didn't mean something else? I mean, in rings like $\displaystyle \mathbb{Z}[\sqrt{-5}]$, there CAN be interesting questions of this sort in such rings.... but this one clearly is not.

4. Originally Posted by topspin1617
Yeah.... are you sure you didn't mean something else? I mean, in rings like $\displaystyle \mathbb{Z}[\sqrt{-5}]$, there CAN be interesting questions

Right, for example $\displaystyle 3,2+\sqrt{-5},2-\sqrt{-5}$ are irreducible elements in $\displaystyle \mathbb{Z}[\sqrt{-5}]$ and $\displaystyle 9=3\cdot 3=(2+\sqrt{-5})\cdot (2-\sqrt{-5})$ so, $\displaystyle \mathbb{Z}[\sqrt{-5}]$ is not a factorial ring. This example provided by Dedekind has had great importance in the foundation of ideals theory to supply the factorization of elements in not factorial rings.