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I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ...
I need some help with understanding Example 1.4.1 ...
Example 1.4.1 reads as follows:View attachment 6516
In the above text by Alaca and Williams we read the following:
"... ... From the first of these, as $$2$$ is irreducible in $$\mathbb{Z} + \mathbb{Z} \sqrt{ -5 }$$, it must be the case that $$\alpha \sim 1$$ or $$\alpha \sim 2$$. ... ...
My question is as follows ... how does $$2$$ being irreducible imply that $$\alpha \sim 1$$ or $$\alpha \sim 2$$. ... ...?
Hope someone can help ...
Peter============================================================================NOTEThe notation $$\alpha \sim 1$$ is Alaca and Williams notation for $$\alpha$$ and $$1$$ being associates ...
Alaca's and Williams' definition of and properties of associates in an integral domain are as follows:https://www.physicsforums.com/attachments/6517
I need some help with understanding Example 1.4.1 ...
Example 1.4.1 reads as follows:View attachment 6516
In the above text by Alaca and Williams we read the following:
"... ... From the first of these, as $$2$$ is irreducible in $$\mathbb{Z} + \mathbb{Z} \sqrt{ -5 }$$, it must be the case that $$\alpha \sim 1$$ or $$\alpha \sim 2$$. ... ...
My question is as follows ... how does $$2$$ being irreducible imply that $$\alpha \sim 1$$ or $$\alpha \sim 2$$. ... ...?
Hope someone can help ...
Peter============================================================================NOTEThe notation $$\alpha \sim 1$$ is Alaca and Williams notation for $$\alpha$$ and $$1$$ being associates ...
Alaca's and Williams' definition of and properties of associates in an integral domain are as follows:https://www.physicsforums.com/attachments/6517