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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:

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

================================================================================

NOTE

The 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:

I need some help with understanding Example 1.4.1 ...

Example 1.4.1 reads as follows:

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

================================================================================

NOTE

The 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: