Norm and square of the ideal $(2,1+\sqrt{-5})$ in the ring of integers

Re your first question....

By choosing a $\mathbb{Z}$-basis $\{ 1, 1 + \sqrt{-5} \}$ for $\mathcal{O}_K$, we can write algebraic integers as coordinate "vectors".

The generators of the ideal $I$ are, relative to this basis, $(2,0)$ and $(0, 1)$. This turns out to be a $\mathbb{Z}$-basis for $I$, although you might want to double check by throwing in all four elements of $\{ 2, 1 + \sqrt{-5} \} \cdot \{ 1, \sqrt{-5} \}$ into a matrix and doing (integer) row reduction to simplify. (using all four of these elements ensures that their span is closed under multiplication by elements of $\mathcal{O}_K$).

One definition of the norm of an ideal is that it is the size of the quotient ring $\mathcal{O}_K / I$. Knowing that the basis for $I$ (relative to the basis for $\mathcal{O}_K$) has coordinate matrix

$$ \left( \begin{matrix}2 & 0 \\ 0 & 1 \end{matrix} \right)$$

makes it easy to see that the quotient group has two elements.

Actually, when we do things this way, we can obtain the norm of $I$ as the determinant of its basis matrix.

As has already been said in the comments by yoyo, $N(I)=[\mathcal{O}_K:I]=2$. One way of thinking about this is that if you multiply any element of $\mathcal{O}_K$ by $2$, then it will always land in $I$, and you can see this by looking at the $\mathbb{Z}$-basis and the generators of $i$.

To compute $I^2$, you just simply multiply the generators together. So here:

• $2\cdot 2=4$
• $2\cdot(1+\sqrt{-5})=2+2\sqrt{-5}$
• $(1+\sqrt{-5})\cdot 2=2+2\sqrt{-5}$ again, so we only need include this in our list of generators once
• $(1+\sqrt{-5})\cdot(1+\sqrt{-5})=-4+2\sqrt{-5}$

This gives us all the generators of the ideal, and

$$ I^2=(2,1+\sqrt{-5})(2,1+\sqrt{-5})=(4,2+2\sqrt{-5},-4+2\sqrt{-5}) $$

About the second point:
In fact, the class number of $\mathbb Q(\sqrt{-5})$ is $2$, so that $I^2$ is principal, whatever the ideal $I$ is. Moreover, by Kummer theory, or Dedekind(?), we know that prime $p$ splits in $\mathbb Q(\theta)$ according to how $f(x)$ factors modulo $p$, where $f$ is the minimal polynomial of $\theta$ over $\mathbb Q$. Now, the minimal polynomial of $\sqrt{-5}$ is $x^2+5\equiv (x+1)^2\pmod2$, so $(2,1+\sqrt{-5})^2=(2)$ by the above-mentioned theory.
Please tell me if some errors occur. Thanks.