sniffnoy | Yay for heavy machinery

Or, another story of Harry doing a problem in a way that he thinks was probably not the intended way, because the way he did it is really complicated.

ADDENDUM that afternoon: Yeah, so you can also do it by noticing it has no real roots, then attempting to factor it into two quadratics, find that the resulting equations require one of three things to be a suqare in Q(√21), and, looking at the resulting equations in Q, none of them are. So indeed, what I did was way more complicated than necessary.

ADDENDUM that night: In fact, my original way doesn't actually work! 3 can't occur as a norm, but -3 can, so 3 isn't prime after all! I realized something was wrong when I noticed, "Wait, 3 divides the discriminant, it shouldn't stay prime!" It won't even be squarefree (and a manual check, to avoid using things I don't currently really understand, shows it's not), which really shoots down Eisenstein.

I don't know if I've said this here before - I've been reading through the Galois Theory section of Dummit & Foote, because when we actually did Galois Theory back in Algebra, I wasn't taking good care of my sleep cycle and I missed a lot of class and ended up basically not learning it.

Anyway so doing problems from there, one of them has a part which ends up reducing to "show that x⁴+3x+3 is irreducible over Q(√21)".

Now how the hell can I do that? If I had any idea what the roots of this thing in C were, I could check all the ways of factoring it and show none are in Q(√21)[x], but, well, I don't want to do that. I certainly don't feel like learning how to solve a quartic right then.

Thought: If Z[(1+√21)/2] is a UFD, it would suffice to show irreducibility over this UFD, and if 3 were prime in that UFD, it would be Eisenstein and we'd be done.

And so a bit of calculation mod 21 shows that 3 cannot occur as a norm in this ring, and so must be irreducible. But is it a UFD at all?

Well, Sloane says it is, but yikes! How to prove this? After putting this aside for a while and working on other things, and then failing to prove it's Euclidean[0] (no idea whether it is or not), finally I remember: Oh, right! Dedekind-Hasse! And since if it is a PID the ordinary norm must be Dedekind-Hasse, I don't have to worry about finding some weird one. And I would go about proving Dedekind-Hasse condition... how?

Well fortunately as an illustration the book has an example with -19. And so I just imitate this with 21 instead, and there's only one case that doesn't go through, after a few minutes I solve that case too.

So Z[(1+√21)/2] is a PID, and 3 is a prime in this ring, and so Eisenstein says we're done.

Somehow, I get the idea that's not how I was supposed to do it.

-Harry

[0]Meaning here Euclidean under the ordinary norm, not just Euclidean in general, which I wouldn't know how to approach.