## Criterion for embedding in cyclic field

$\mathbf{Problem:}$ Assume $F$ has $p$ distinct pth roots of 1. $p$ a prime, and $E/F$ is cyclic of dimensional $p^f$. Let $z$ be a primitive $p$th root of 1. Show that if $E/F$ can be imbedded in a cyclic field $K/F$ of dimension $p^{f+1}$, then $z=N_{E/F}(u)$ for some $u\in E$.

$\mathbf{Proof:}$ Suppose $\sigma$ is the generating isomorphism of cyclic Galois group $K/F$. Then $\displaystyle E=\text{Inv }\sigma^{p^f}$. $N_{K/E}(z)=z^p=1$, so by Hilbert satz 90, there exists $a\in K$ such that $\displaystyle z=\sigma^{p^f}(a)a^{-1}$ .
$\displaystyle \sigma^{p^f}(a^{-1}\sigma(a))=\sigma^{p^f}(a^{-1})\sigma^{p^f+1}(a)=(z a)^{-1}\sigma(z a)=a^{-1}\sigma(a)$.
So $a^{-1}\sigma(a)=a_0\in E$, then
$N_{E/F}(a_0)=a^{-1}\sigma(a)\sigma(a^{-1})\sigma^2(a)\cdots \sigma^{p^f-1}(a)\sigma^{p^f}(a)=a^{-1}\sigma^{p^f}(a)=z$.

$\text{Q.E.D}\hfill \square$

$\mathbf{Remark:}$ Jacobson p300.