Urgent: Number Theory-Wilson's Theorem

[mathsss2]

Urgent:

1. let p be prime form 4k+3 and let a be an integer. Prove that a has order p-1 in the group U(\frac{\texbb{Z}}{p\texbb{Z}}) iff -a has order \frac{(p-1)}{2}

2. let p be odd prime explain why: 2*4*...*(p-1)\equiv (2-p)(4-p)*...*(p-1-p)\equiv(-1)^{(p-1)/2}*1*3*...*(p-2)mod p.

3. Using number 2 and wilson's thereom [(p-1)!\equiv-1 mod p] prove 1^23^25^2*...*(p-2)^2\equiv(-1)^{(p-1)/2} mod p

Thanks.

 

[robert Ihnot]

For p=3+4k, then (p-1)/2 is an odd number. Thus 1\equiv (-a)^\frac{p-1}{2} =-a^\frac{p-1}{2}

Consequently the right side would be -1 under the circumstances that both powers of a equalled 1.

 

[mathsss2]

I got the rest, I still need help in #2.