Finding the minimal polynomial of a polynomial in $n$ variables over a field adjoined by the elementary symmetric polynomials

Question

I would like to know how you might generally try to find the minimal polynomial so that I can try to do it for a specific case.

The elementary definitions in $n$ variables are defined to be:

$$s_1 = X_1 + \dots + X_n$$ $$s_2 = \sum_{i<j}X_iX_j$$ $$ \vdots$$ $$s_n = X_1\dots X_n$$

Now let $K$ be a field and let $M = K(s_1, \dots, s_n)$. How might you begin to find the minimal polynomials of various polynomials in $K[X_1, \dots, X_n]$? For example, $X_1X_2$?

If $n$ were to equal $2$ then it would simply be the linear polynomial $f(t) = t - s_2 = t - X_1X_2$, but what could we do if, say, $n=3$?

The specific case I want to try to work out is $X_1X_3 + X_2X_4$ in the case where $n=4$, in case there is anything special about it that suggests there may be a slightly rogue way of approaching it.

Overall, I would like to know if there are any techniques for finding the minimal polynomials for these multivariate polynomials over this field over symmetric polynomials.

It is a lot less intuitive than when working over fields such as $\mathbb Q$ since I can just keep applying standard operations until the value ends up in $\mathbb Q$, but in this case I am really struggling to keep track of all the symbols and can't help but wonder if there would be an easier way to do this.

Answer 1

Let $L$ be a finite Galois extension of $K$ with Galois group $G$, and let $a \in L$ be any element.

Exercise #1: The minimal polynomial of $a$ over $K$ is the product $m(t) = \prod_{a' \in \text{Orb}(a)} (t - a')$ where $\text{Orb}(a) = \{ ga : g \in G \}$ is the orbit of $a$ under the action of $G$.

Exercise #2: $K(x_1, \dots x_n)$ is a finite Galois extension of $L(s_1, \dots s_n)$ with Galois group $S_n$ (acting by permutation on the $x_i$).

(For the purposes of this calculation the definition of "finite Galois extension" you should use is that $G$ is a finite group acting on $L$ such that the fixed field $L^G = \{ \ell \in L : g \ell = \ell \forall g \in G \}$ is $K$, since this is all we need.)

This completely solves your problem. For example, when $a = x_1 x_2$ the orbit of $a$ under permutation consists of all polynomials of the form $x_i x_j$ where $i < j$ (this is to avoid duplicates, since $x_1 x_3 = x_3 x_1$ and so forth), hence the minimal polynomial has degree ${n \choose 2}$ and is

$$\prod_{i < j} (t - x_i x_j).$$

The exercises imply that the minimal polynomial always has degree the size of the orbit under permutation, which generically has size $n!$ and in general divides $n!$.