Calculating the norm of an ideal in Z[√6]

[Firepanda]

For part i) I deduced via Dedekind's criterion that

<2> = <2,√6>² & <3> = <3,√6>²

So ii) I am trying to do now, and my argument is thus:

Let a be an ideal in Z[√6]. Suppose that N(a) = 24.

By a proposition in my notes we have that

a|<24> = <2,√6>⁶<3,√6>²

so a = <2,√6>^r<3,√6>^s

for some r in {0,1,2,3,4,5,6} and s in {0,1,2}

We have N(a) = N(<2,√6>)^rN(<3,√6>)^s, and from that I can deduce what r and s should be.

So now I need to calculate the norms of these two ideals.

How do I calculate the norm of <2,√6>? Can I find a Z-basis for this ideal so that I can find a base change matrix and find the norm of that?

Thanks

 

[morphism]

In fact <2> and <3> are principal in Z[sqrt{6}].

I'll explain why this is the case for <2>. First off, because <2>=<2,√6>^2, it suffices to show that <2,√6> is principal. For this, it suffices to exhibit an element of norm +/-2 (why?). So all we need to do is find a,b such that a^2-6b^2=+/-2, but this is pretty easy.