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T-O7
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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) a solves [tex]x^2+x=-1(mod p)[/tex] if and only if a is 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
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) a solves [tex]x^2+x=-1(mod p)[/tex] if and only if a is 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