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.