# How to multiply these two prime ideals in Z[√(-5)]? [duplicate]

How to calculate the product of the below multiplication of prime ideals in Z[√(-5)]?

(2, 1-√(-5))(3, 1+√(-5))

I know it can be firstly expressed as a non-principal ideal generated by three numbers in Z[√(-5)],

(6, 2+2√(-5), 3-3√(-5))

but I don't know if it can be further simplified (and how, if it can).

Thanks for your help!

• Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. Commented Apr 9, 2023 at 7:52
• Please check out the abridged guide to new askers. It is difficult to give a helpful answer to your question, because your background is in the dark. You do hint at knowing this ring is NOT a principal ideal domain (making your choice of tag very strange). But how familiar are you with manipulating ideals of rings of integers of number fields? In this ring the ideals are necessarily free abelian groups of rank two, so how could you possibly need more than two generators. Also, do search for other questions about this ring on the site. Commented Apr 9, 2023 at 8:01
• The background of my inquiry is, in my understanding, cross-multiplication of prime ideals shall correspond to prime factorization. In this case, (2,1+√(-5))(2,1-√(-5))=(2), (3,1+√(-5))(3,1-√(-5))=(3); (2,1+√(-5))(3,1+√(-5))=(1+√(-5)), (2,1-√(-5))(3,1-√(-5))=(1-√(-5)); these two pairs correspond to the two ways of prime factorization of 6 in Z[√(-5)]. However, I cannot figure out how to interpret the remaining possible cross-multiplication, namely (2,1-√(-5))(3,1+√(-5)) and (2,1+√(-5))(3,1-√(-5)). Commented Apr 9, 2023 at 8:11
• This thread has a lot of example calculations on how to manipulate ideals of this ring. Commented Apr 9, 2023 at 8:12
• Looking at this ideal a subgroup of $\Bbb{Z}[\sqrt{-5}]$, using the integral basis $\{1,\sqrt{-5}\}$, we see that the generators have coordinate vectors $(6,0),(2,2),(3,-3)$ so the vector $(6,0)-(2,2)-(3,-3)=(1,1)$ is in there. The ideal has norm six, so sure looks like it must be the principal ideal $(1+\sqrt{-5})$, doesn't it? Anyway, because the class group has size two, it follows that the product of two non-principal ideals is principal. Commented Apr 9, 2023 at 8:20