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On an exam for my modern algebra class, we were asked to determine whether or not $\Bbb{Q}\left(\alpha\right)$ is the splitting field (here using the convention that splitting field means the smallest field such that $p$ factors into linear factors) for the minimal polynomial of $\alpha = \sqrt{3 - \sqrt{6}}$ over $\Bbb{Q}$. I know that the answer is no: the minimal polynomial for $\alpha$ is $$ p(x) = x^4 - 6x^2 + 3 = (x - \alpha)(x + \alpha)(x - \beta)(x + \beta), $$ where $\beta = \sqrt{3 + \sqrt{6}}$. If $\Bbb{Q}(\alpha)$ were the splitting field for $p$, then $\beta\in\Bbb{Q}(\alpha)$, which would imply that $\alpha\beta = \sqrt{3}\in\Bbb{Q}(\alpha)$, which further implies that $\sqrt{2}$ would have to be in $\Bbb{Q}$. However, we can see that this is impossible by seeing that $a + b\alpha + c\sqrt{6} + d\sqrt{6}\alpha = \sqrt{2}$ has no solutions with $a,b,c,d\in\Bbb{Q}$ by squaring and taking cases.

This method works, but it is rather uninspiring and tedious. My question: is there a better way to see this? It appears that in general, we don't really have terribly powerful tools for determining splitting fields, their degrees, and linear dependence/independence of roots of a polynomial. We do know that if $E$ is a splitting field for $p$ over $F$, then $$ \left[E : F\right] \leq \deg p! $$ This bound provides a limit on the amount of possibilities for the splitting field, but it may or may not be useful (it wasn't in my case). After all, the main problem for the case of $\Bbb{Q}(\alpha)$ was determining whether or not $\beta\in\Bbb{Q}(\alpha)$. So, given a polynomial $p\in F[x]$ that splits in an extension $E$ of $F$ with roots $r_1,\ldots, r_n$, are there other useful results or clever methods for computing how many and/or which roots one must adjoin to $F$ to obtain a bonafide splitting field $F\left(r_{i_1},r_{i_2},\ldots,r_{i_k}\right)$ for $p$? Of course, what methods need to be used will vary from polynomial to polynomial, but I'm looking for some relatively general strategies that don't involve ugly systems of nonlinear equations (for example, determining if $F\left(r_{i_1},r_{i_2},\ldots,r_{i_k}\right)$ is a splitting field using some sort of degree considerations). If no such strategies exist, I'd also be interested in seeing a few computations of splitting fields that employ different types of arguments.

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  • $\begingroup$ I'm not sure what text you are using, but you may enjoy D. Cox's "Galois Theory" (specifically Chapters 5 and 6). $\endgroup$ – Benjamin Dickman Apr 6 '13 at 4:41
  • $\begingroup$ There are many tools provided by Galois theory. They often appear to be ad hoc (at least to a non-expert like me), but do settle the low degree cases. You didn't list Galois theory among your assets? Is your course heading in that direction? The tools that emerge will not be as generic as one might hope, but I cannot give really useful examples without using those. $\endgroup$ – Jyrki Lahtonen Apr 6 '13 at 7:04
  • $\begingroup$ @JyrkiLahtonen My course has just stated the fundamental theorem of Galois theory, but I am familiar with Galois theory from my other studies. I would be perfectly happy with answers using results from Galois theory; I didn't mean to suggest that answers should avoid using Galois theory, I just didn't see any application of Galois theory to the problem I had in mind when I posed the question. $\endgroup$ – Stahl Apr 6 '13 at 14:51

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