User:LDH

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.

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

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

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

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

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

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

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