# Primes in ring of Gauss integers - help

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 $$R=\field{Z}[\zeta]$$ where
$$\zeta=\frac{1}{2}(-1+\sqrt{-3})$$
is a cube root of 1.
Then the question is:
Show the polynomial $$x^2+x+1$$ has a root in $$F_p$$ if and only if $$p\equiv1 (mod 3)$$.

I thought i could show this in two steps, by showing that:
a) a solves $$x^2+x=-1(mod p)$$ if and only if a is an element of order 3 in $$F^x_p$$.
b)$$F^x_p$$ contains an element of order 3 if and only if $$p\equiv1 (mod 3)$$.

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

## Answers and Replies

Hurkyl
Staff Emeritus
Gold Member
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)

Hmm....sorry, i don't see what you mean.

Hurkyl
Staff Emeritus
Gold Member
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)

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!
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.

Hurkyl
Staff Emeritus
Gold Member
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.

I was looking at a somewhat similar problem, https://www.physicsforums.com/showthread.php?t=60863

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

Example: $$7=2^2+2+1=(2-\zeta)(2-\zeta^2)$$

By the way, as Hurkyl points out, this is an algebratic number theory problem.

Last edited:
mathwonk