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. $\mathfrak{b}=\left<3,1+\sqrt{-5}\right>$)?
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.