Primes in ring of Gauss integers - help

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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 :cry:
 

Answers and Replies

  • #2
Hurkyl
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Well, first I think you need to review the problem and definitions... the things you've said don't seem to connect to one another.


Anyways, I think this might help: note that

x^3 - 1 = (x - 1) (x^2 + x + 1)
 
  • #3
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Hmm....sorry, i don't see what you mean.
 
  • #4
Hurkyl
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It means that if a is a root of x^2 + x + 1, then it is also a root of x^3 - 1. (i.e. it is a cube root of 1)
 
  • #5
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ah, so a^3=1, and a has order 3. We can apply the argument backwards, and that will prove a). I see, thanks Hurkyl! :smile:
There's also a second part to this problem, which says (p) is prime ideal in R if and only if p=-1 (mod 3)
Apparently the first part of this problem applies, but i'll have to think about this more.
 
  • #6
Hurkyl
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Ah, so that's why you mentioned R.

(Incidentally, I think you meant algebraic integers, not Gaussian integers)

I don't know if it will help, but note that if Fp has a root of x^2 + x + 1, then there is a homomorphism from R onto Fp.
 
  • #7
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I was looking at a somewhat similar problem, https://www.physicsforums.com/showthread.php?t=60863

You use quadratic reciprocity on [tex] X^2\equiv-3 [/tex] Mod p to discover that p is of the form 3k+1. Thus, this prime splits over a field with[tex] \sqrt{-3}, [/tex]so it would not generate a prime ideal.

Example: [tex]7=2^2+2+1=(2-\zeta)(2-\zeta^2) [/tex]

By the way, as Hurkyl points out, this is an algebratic number theory problem.
 
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  • #8
mathwonk
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this is elementary. look at the group homomorphism from the multiplicative group Fp - {0} to itself defined by cubing. then if there is a primitive cube root of 1, the map is 3 to 1, and has image of order 1/3 the order of the group, i.e. then 3 divides p-1. on the other hand if 3 divides the order of the group, it is elementary group theory that there exists an element of order 3.
 

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