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Homework Help: Urgent: Number Theory-Wilson's Theorem

  1. Nov 2, 2008 #1

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

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

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

    Last edited: Nov 3, 2008
  2. jcsd
  3. Nov 2, 2008 #2
    For p=3+4k, then (p-1)/2 is an odd number. Thus [tex]1\equiv (-a)^\frac{p-1}{2} =-a^\frac{p-1}{2} [/tex]

    Consequently the right side would be -1 under the circumstances that both powers of a equalled 1.
    Last edited: Nov 2, 2008
  4. Nov 3, 2008 #3
    I got the rest, I still need help in #2.
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