larryhammick at telus dot net

At Wiki I do a little on mathematics and a fair amount on Islamist, mostly Sunni, terrorism.

Below is my official secret Testing Ground.

A circular permutation on a set of k elements has sign ${\displaystyle (-1)^{k-1}}$. Let i be the order of m in ${\displaystyle U_{p}}$. The permutation ${\displaystyle \tau _{m}}$ consists of ${\displaystyle (p-1)/i}$ orbits, each of size i, whence

${\displaystyle \epsilon (\tau _{m})=(-1)^{(i-1)(p-1)/i}}$

If i is even then ${\displaystyle m^{i/2}=-1}$ and so

${\displaystyle m^{(p-1)/2}=m^{{\frac {i}{2}}{\frac {p-1}{i}}}=(-1)^{(p-1)/i}=\epsilon (\tau _{m})}$

If i is odd then 2i divides p-1, so

${\displaystyle m^{(p-1)/2}=m^{i{\frac {p-1}{2i}}}=1=\epsilon (\tau _{m})}$

In both cases, Zolotarev's lemma follows from Euler's Criterion.