A note on the product of the conjugates of a polynomial

TL;DR

This paper clarifies a misconception about the product of conjugates of a polynomial dividing a separable irreducible polynomial over a field, showing it equals a power of the original polynomial under certain conditions.

Contribution

It precisely characterizes when the product of conjugates of a polynomial dividing a separable irreducible polynomial equals a power of that polynomial.

Findings

The product of conjugates of g over K[X] equals f^n for some n.

The misconception about the product being always f is incorrect.

The relation depends on the coefficient field of g and the Galois extension.

Abstract

The theorem proved in this note, although elementary, is related to a certain misconception. If $K$ is a field, $f\in K[X]$ is separable and irreducible over $K$, and $g$ is a polynomial dividing $f$, whose coefficients lie in some finite Galois extension of $K$, it may seem natural to assert that the product of the conjugates of $g$ over $K[X]$ is $f$. But this assertion is wrong, except in one particular case. In this note, we make the relation between $K$, $f$, the product of the conjugates of $g$, and the coefficient field of $g$, precise. In particular, it is shown that the product of the conjugates of $g$ over $K[X]$ is equal to $f^n$, with $n\in \mathbb N$.