When Does F(α^d) = F(α)?

Abstract

Suppose that α is algebraic over a field $\mathbf{F}$. A standard exercise in a first course in field theory is to show that if the degree of the minimal polynomial of α over $\mathbf{F}$ is odd, then $\mathbf{F}({\alpha }^2)=\mathbf{F}(\alpha )$. In this note, we generalize the sufficient conditions on the minimal polynomial for α so that $\mathbf{F}({\alpha }^d)=\mathbf{F}(\alpha )$ for any particular integer $d\ge 2$. Then, given any finite set $\mathcal{D}$ of integers $d\ge 2$, this generalization allows us to construct irreducible polynomials f, with $f(\alpha )=0$, such that $\mathbf{F}({\alpha }^d)=\mathbf{F}(\alpha )$ for all $d\in \mathcal{D}$.

MSC::

• Primary 12F05
• Secondary 11G25

Acknowledgment

The author thanks the anonymous referees for their valuable suggestions.