## Albert radicals of norm and Hilbert’s Satz 90

$\mathbf{Problem(Albert):}$ Let $E$ be a cyclic extension of dimension $n$ over $F$ and let $\eta$ 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 $E/F$ of dimensionality $m$ such that $c=N_{K/F}(v)$.

$\mathbf{Proof:}$ $G=\text{Gal }E/F$ is a cyclic group, then it  has a unique subgroup $H=\{\eta^m,\eta^{2m},\cdots,\eta^{rm}=1\}$ of index $r$. By the Galois corresponding theorem, there exists a unique subfield $K=\text{Inv }H$ such that $\text{Gal K/F}\cong G/H$. $\text{Gal }K/F=\{\eta^1|_K,\eta^2|_K\cdots,\eta^m|_K\}$. $K$ has dimensionality $m$ over $F$.

Consider $w=c^{-1}u\eta(u)\eta^2(u)\cdots \eta^{m-1}(u)$, then $\displaystyle\eta(w)=\frac{\eta^m(u)}{u}w$. We also have

$\displaystyle N_{E/K}(w)=\eta^m(w)\eta^{2m}(w)\cdots \eta^{rm}(w)=c^{-r}\eta(u)\eta^2(u)\cdots\eta^n(u)=1$,

by Hilbert’s Satz 90, $\exists \, l\in E$ such that $\displaystyle w=\frac{\eta^m(l)}{l}$.

Let $\displaystyle v=\frac{ul}{\eta(l)}$, then $v\in K$, because

$\displaystyle \eta^m(v)=\frac{\eta^m(u)\eta^m(l)}{\eta^{m+1}(l)}=\frac{\eta^m(u)\eta(l)}{\eta^{m+1}(l)}\cdot\frac{\eta^m(l)}{\eta(l)}=\frac{\eta^m(u)}{\eta(w)}\cdot\frac{\eta^m(l)}{\eta(l)}=\frac{\eta^m(u)}{\eta(w)}\cdot\frac{wl}{\eta(l)}=u\frac{l}{\eta(l)}=v$

Surprisingly we have

$\displaystyle N_{K/F}(v)=v\eta(v)\cdots \eta^{m-1}(v)=u\eta(u)\cdots\eta^{m-1}(u)\frac{l}{\eta^m(l)}=cw\frac{l}{\eta^m(l)}=c$.

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

$\mathbf{Remark:}$ Jacabson p300. This problem puzzled me for three weeks. Finally it turns out to be very easy.

• Mert Sevinis  On December 13, 2012 at 7:46 pm

A similar question arises as follows

$K, F$ be cyclic extensions of $k$ with $[K:k]= [F:k]=p$. Let $k\subset L\subset KF$ with $[L:k]=p$. Let $a\in k$ and suppose that $a=N_{K/k}(b) = N_{F/k}(c)$ where $b\in K$ and $c\in F$. Show that there is an element $d\in L$ such that $a=N_{L/K}(d)$.

I think I have an idea about the proof; I just write all elements explicitly and at the end I obtain $p$ equations with $p$ unknowns. However, I don’t use Hilbert 90, and the question seems to ask an answer using Hilbert 90. A proof using Hilbert 90, is it always tricky and does it come out by surprise, or is there a theory behind it?

By the way thank you for this answer, I gained very much of time.

• Mert Sevinis  On December 15, 2012 at 8:50 pm

I think we may erase this comment (including this one:)), because it has flaws. Instead, if I’m not mistaken, the question has a similar answer: (sketch) Let $\sigma, \tau$ be extensions of the generators of $G(K/k)$ and $G(F/k)$ respectively. For some $\eta = \sigma^{i_0}\tau^{j_0}$ $F()=L$. $N_{KF/L}(b^{-1}c)=1$, by Hilbert 90 $\exists l\in KF b^{-1}c = l^{-1}\eta(l)$. We also take $G(L/k) =$, and then set $d = c l/\sigma^{i_0}(l)$.

• Sun's World  On December 15, 2012 at 10:22 pm

Hehe. I almost forgot galois theory. This is the course I learned last semester.