I'm working through an old paper in preparation for an upcoming exam, and this question is troubling me.
Let $\vartheta$ be an algebraic number such that $\vartheta^{7} = 12$, and let $K = \mathbb{Q}(\vartheta)$. Determine the ring of integers of $K$.
I was unable to do two of the preceding questions as I've not been taught the required material. These were:
- What is the discriminant of $\mathbb{Z}[\vartheta]$?
- Prove that the order $\mathbb{Z}[\vartheta]$ is $3$-maximal and $7$-maximal.
I have not been taught the discriminant of $\mathbb{Z}[\vartheta]$ for some algebraic number, nor the notion of $n$-maximal, so I skipped these. The other question was to show that $\varsigma = \vartheta^{4}/2$ is an algebraic integer.
I'm very aware that such a question is unlikely to come up in my exam, as it seems I haven't been taught the right material to answer it.
However, I thought that I'd attempt the quoted question above with the methods that I have been taught, but I couldn't get anywhere without very lengthy calculations because of the seventh power. We have mostly stuck to quadratic and some cubic fields, for which the integral basis is usually given or there is a theorem to check. Typically, the integral basis would be something of the form $$ 1, \omega, \omega^{2} $$ for a cubic (though I am aware that this is not always the case), but I feel it would be wrong to guess an integral basis for $\mathbb{Z}[\vartheta]$ is $$ 1, \vartheta, \vartheta^{2}, \vartheta^{3}, \vartheta^{4}, \vartheta^{5}, \vartheta^{6}. $$
I couldn't find anything online for dealing with number fields of degree seven, but I wondered if there was an 'obvious' integral basis that one would guess from the questions that came before it?
Well, for this you really have to know about discriminants. I’ll try to give you some guidance.
If $R$ is a ring that’s free as a $\Bbb Z$-module, with basis $\{b_1,\cdots,b|n\}$, then the discriminant may be defined as the determinant of the trace pairing. That is, you take the matrix whose $(i,j)$ entry is $\text{Tr}^R_{\Bbb Z}(a_ia_j)$, and find its discriminant. That trace is the ordinary trace of Galois theory. Fortunately, when $R=\Bbb Z[\theta]$, there’s a far quicker way to calculate the discriminant of $R$: If $f$ is the minimal polynomial for $\theta$ over $\Bbb Q$, then up to sign, the discriminant is just $\mathbf N^R_{\Bbb Z}\,f'(\theta)$.
What’s it good for? The discriminant contains all the primes of $\Bbb Z$ that ramify in the extension.
Now, something you should really prove for yourself is that when $R\supset R'$, with the index of $R'$ in $R$ being $m$, then $\text{disc}(R'/\Bbb Z)=m^2\text{disc}(R/\Bbb Z)$. In other words, if $R'$ is smaller, its discriminant is too big, and by a square factor.
Consequence: If you ever get a ring of integers in a number field $K$ whose discriminant is square-free, you know automatically that it’s the ring of all $K$-integers.