cubic residue

• Apr 6th 2013, 12:04 PM
wingtip
cubic residue
p is an odd prime, and p does not divide u prove that if p is congruent to 2 mod 3, then every unit is a cubic residue mod p. I don't know where to start with this.
• Apr 6th 2013, 02:14 PM
HallsofIvy
Re: cubic residue
Well, you might start with the definition of "cubic residue". What is that?
• Apr 6th 2013, 02:45 PM
wingtip
Re: cubic residue
u is a cubic residue mod p if there exists some b such that u is congruent to b^3 mod p
• Apr 6th 2013, 05:13 PM
johng
Re: cubic residue
Hi,
If you know a few facts, this is easy. Otherwise, it might be kinda hard.
The multiplicative group of $\displaystyle \mathbb Z_p$ is cyclic with generator, say x (a primitive root mod p). Then the order of $\displaystyle x^3$ is the number of elements in the set $\displaystyle \{x^{3k}\,:\, k\in\mathbb Z\}={p-1\over gcd(p-1,3)}$. Now if 3 divides p - 1, then p is congruent to 1 mod p, but p is congruent to 2 mod 3. So gcd(p-1,3)=1. That is, the order of $\displaystyle x^3$ is p - 1 and so every non-zero element of $\displaystyle \mathbb Z_p$ is a cube; i.e. a cubic residue.