(adsbygoogle = window.adsbygoogle || []).push({}); Primes in ring of Gauss integers - help!!

I'm having a very difficult time solving this question, please help!

So i'm dealing with the ring [tex]R=\field{Z}[\zeta][/tex] where

[tex]\zeta=\frac{1}{2}(-1+\sqrt{-3})[/tex]

is a cube root of 1.

Then the question is:

Show the polynomial [tex]x^2+x+1[/tex] has a root in [tex]F_p[/tex] if and only if [tex]p\equiv1 (mod 3)[/tex].

I thought i could show this in two steps, by showing that:

a)asolves [tex]x^2+x=-1(mod p)[/tex] if and only ifais an element of order 3 in [tex]F^x_p[/tex].

b)[tex]F^x_p[/tex] contains an element of order 3 if and only if [tex]p\equiv1 (mod 3)[/tex].

I've proved part b, but i can't seem to get a hold of a.

Please help

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# Primes in ring of Gauss integers - help

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