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 pth 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.

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