On the order and degree of solutions to pure equations

Abstract:

Let $K$ be a field. Let $\theta$ be an element of a field extension of $K$. The order of $\theta$ over $K$ is the smallest positive integer $m$ such that ${\theta ^m}$ lies in $K$, or $\infty$. We compare the order $m$ of $\theta$ to the degree $h$ of $\theta$ over $K$. Clearly $h \leqslant m$. Theorem. Let $K$ be a field. Let $\theta$ be an element of degree $h$ and order $m$ over $K$. Let $p$ be a prime. Let ${p^e}$ be the maximum power of $p$ dividing $h$, and suppose ${p^s}$ divides $m$. (1) If the characteristic of $K$ is $p$, then $s \leqslant e$. (2) If $s > e$ and $p$ is odd, then $K(\theta )$ contains a primitive $p$th root of unity $u$ not in $K$. Moreover $K(u)$ contains a primitive ${p^{s - e}}$ root of unity. (3) If $s > e$ and $p = 2$, then $- 1$ is not a square in $K$ and $K(\theta )$ contains $i = \sqrt { - 1}$. Moreover $- 1$ is a ${2^{s - e}}$ power in $K(i)$.