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.

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.