Minimal degree of an element of a number field with respect to its quadratic extension

Abstract

In this paper, we show that every quadratic algebraic number lying in a quartic extension L of ${\mathbb {Q}}$ can be written as a quadratic polynomial with rational coefficients in a generator of L over ${\mathbb {Q}}$. On the other hand, we prove that for each totally real algebraic number $\beta$ of degree $d \ge 3$, there are infinitely many quadratic extensions L of ${\mathbb {Q}}(\beta )$ such that $\beta$ cannot be expressed by a quadratic polynomial with rational coefficients in a generator of L over ${\mathbb {Q}}$. The same assertion is proved for a large class of cubic algebraic numbers $\beta$ without assumption that $\beta$ is totally real.