I'm trying to calculate the class group of $\mathbb{Q} (\sqrt{-21})$, and I've managed to confuse myself. I've used the standard Minkowski bound to show that we only need to consider principal ideals (p) for p $\leq$ 5, and have found that (2) = $\mathcal{P}^2$, (3) = $\mathcal{Q}^2$, (5) = $\mathcal{R}_1 \mathcal{R}_2$ for $\mathcal{R}_1 \neq \mathcal{R}_2$.
Then the norm is $N(a + b\sqrt{-21}) = a^2 + 21 b^2$ and we want to find out some more information about our ideals (none of which are principal): take a = 2 and b = 1: then $(2 + \sqrt{-21}) = \mathcal{R}_1 \mathcal{R}_1$ WLOG (since 5 doesn't divide $2^2 + 1^2 \sqrt{-21}$, it is divisible by just one of the $\mathcal{R}_i$: as in the handout by Keith Conrad - see on the first page) but then $\mathcal{R}_1$ has order 2 and is self inverse, so then $\mathcal{R}_2$ is the inverse so $= \mathcal{R}_1$, so they are not distinct: except they should be. So where have I gone wrong?
I don't know whether or not I've made a logical error, but if not I suspect the problem may be arising because you can factorise 25 in 2 distinct ways. Can anyone help me? I hope this makes sense, I think I have probably left off some square brackets but I hope the general approach I am taking (very similar to Conrad's handout) is comprehensible. Many thanks! -Pete