0

I am trying to understand ideal notation with pointed brackets and how to use it.

For instance, if I had an ideal $\mathfrak{a}=\left<2,1+\sqrt{-5}\right>$, where $2$ and $1+\sqrt{-5}$ are its generators, what does this mean for the format of the ideal?

And how would I find powers of this, i.e. $\mathfrak{a}^2$ or $\mathfrak{ab}$ for $\mathfrak{b}$ an ideal of the same form, (e.g. $\mathfrak{b}=\left<3,1+\sqrt{-5}\right>$)?

AccioHogwarts
  • 1,437

1 Answers1

0

We have $\mathfrak{a}^2=(2^2, 2(1+\sqrt{-5}), 1+2\sqrt{-5}-5)=(2)$ by definition of the ideal product $IJ$ for two ideals in the ring $R=\mathbb{Z}[\sqrt{-5}]$. For discussions on the ideal product see here and here.

Community
  • 1
Dietrich Burde
  • 140,055
  • Thank you, I now understand that multiplying the generators will find the generators of the things in the product. I am still confused, however, as to why $(2^2,2(1+\sqrt{-5}),1+2\sqrt{-5}-5)=(2)$? – AccioHogwarts Mar 24 '15 at 05:01
  • Wait, I think I have it; $2^2, 2(1+\sqrt{-5}),2\sqrt{-5}-4$ are all made by multiplying the elements of $\mathbb Z \left[\sqrt{-5} \right]$ by multiples of $2$ and combining them? Is this right? – AccioHogwarts Mar 24 '15 at 05:08
  • Yes, you are right. – Dietrich Burde Mar 24 '15 at 09:19