Recently I was thinking about images of number field elements under a polynomial with coefficients in a smaller field, and I came across the following construction. It did not have the properties I wanted for the problem I was thinking about, but it seemed natural enough that it might have some useful interpretation or application. I could not come up with anything, so I am asking here.
Suppose we have a tower of finite extensions $L/K/F$ with $K/F$ Galois with group $G = \mathrm{Gal}(K/F)$, and let $f(x) \in K[x]$ be the minimal polynomial of an element $\beta \in L$ over $K$. If we define the polynomial $$f_{\Pi}(x) = \prod_{\sigma \in G} {^{\sigma}}f(x),$$ then now $f_{\Pi}(x) \in F[x]$ and $f$ coincides with the minimal polynomial of $\beta$ over $F$. Here the action is just on the coefficients of the polynomial.
Instead, define the polynomial $$f_{\Sigma}(x) = \sum_{\sigma \in G} {^{\sigma}}f(x).$$ Again $f_{\Sigma} \in F[x]$, now defined over the base field. If you like, use the averaged polynomial $\frac{1}{[K:F]} f_{\Sigma}$ to produce a monic polynomial.
$f_{\Pi}$ is like the norm of $f$ from $K[x]$ to $F[x]$ and is the minimal polynomial over $F$. $f_{\Sigma}$ is like the trace of $f$ from $K[x]$ to $F[x]$, but what is it? Does it have any useful interpretation? Has that construction found use in any applications?
Note: If $L = K(\beta)$, then $\mathrm{deg}(f_{\Pi}) = [L:F]$. And $\mathrm{deg}(f_{\Sigma}) = [L:K]$. This degree property was one of the stipulations I had for the polynomial, and $f_{\Sigma}$ happened to be a way of producing one.
