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Math Help - cubic residue

  1. #1
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    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.
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  2. #2
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    Re: cubic residue

    Well, you might start with the definition of "cubic residue". What is that?
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  3. #3
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    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
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  4. #4
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    Re: cubic residue

    Hi,
    If you know a few facts, this is easy. Otherwise, it might be kinda hard.
    The multiplicative group of \mathbb Z_p is cyclic with generator, say x (a primitive root mod p). Then the order of x^3 is the number of elements in the set \{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 x^3 is p - 1 and so every non-zero element of \mathbb Z_p is a cube; i.e. a cubic residue.
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