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I am stuck by a problem in Jacobson I for a long time. It seems not too difficult. Hope anyone can help me out. It is on page 300.

Let $E$ be a cyclic extension of dimension $n$ over $F$ and $\eta$ be a generator of $\text{Gal }E/F$. Let $r|n$, $n=rm$ and suppose $c$ is a non-zero element of $F$ such that $c^r=N_{E/F}(u)$ for some $u\in E$. Show that there exists a $v$ in the unique subfield $K$ of $\text{Gal }E/F$ of dimensionality $m$ such that $c=N_{K/F}(v)$.

This is a theorem due to Albert. I also checked his book: Structure of Algebra. But I really can not understand what he was saying. Can we solve this problem just by Galois theory? Thank you very much.

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I finally know how to solve this problem. See my page… – Slm2004 Apr 11 '12 at 15:06

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