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I'm still making my way along in Niven's Intro to Number Theory, and the title problem is giving me a little trouble near the end, and I was hoping someone could help get me through it.

Now $x^8\equiv 16\pmod{2}$ is solvable with $x\equiv 0\pmod{2}$, so I assume $p$ is an odd prime. From a theorem earlier in the text,

If $p$ is a prime and $(a,p)=1$, then the congruence $x^n\equiv a\pmod{p}$ has $(n,p-1)$ solutions or no solution according as $a^{(p-1)/(n,p-1)}\equiv 1\pmod{p}$ or not.

So since $(16,p)=1$, the problem reduces to showing that $16^{(p-1)/(8,p-1)}\equiv 1\pmod{p}$ holds for all $p$. I note that $(8,p-1)$ can only take values $2,4,8$. For $2$, the above equivalence is then $4^{p-1}\equiv 1\pmod{p}$, which is true by Fermat's little Theorem. For $4$, it is then $2^{p-1}\equiv 1\pmod{p}$, which again holds by FlT. However, the case where $(8,p-1)=8$ is throwing me off. At best I see that $16^{(p-1)/8}\equiv 2^{(p-1)/2}\pmod{p}$, but I'm not sure how to show this is congruent to $1$ modulo $p$. Maybe there's a more elegant way to do it without looking at cases. Thanks for any insight.

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  • $\begingroup$ This problem naturally splits into cases. Don't be afraid to think casewise. $\endgroup$ – Qiaochu Yuan Oct 14 '10 at 9:56
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    $\begingroup$ This problem played a role in the development of the Grunwald–Wang theorem; see en.wikipedia.org/wiki/Grunwald-Wang_theorem#History $\endgroup$ – lhf Oct 17 '10 at 23:46
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One way is to use the Legendre symbol identity $2^{(p-1)/2} \equiv (\frac{2}{p}) \equiv (-1)^{(p^2-1)/8} \pmod p$ (for odd primes p), keeping in mind that if $(8,p-1)=8$ then $p \equiv 1 \pmod 8$.

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  • $\begingroup$ Thanks, I overlooked that identity. I had a small question to add. I know if $a$ is a quadratic residue modulo $p$, then the Legendre symbol $(a|p)=1$ . However, if by some calculation I find that $(a|p)=1$, is it also the case that $a$ is a quadratic residue modulo $p$? $\endgroup$ – yunone Oct 14 '10 at 10:13
  • $\begingroup$ Yep that's right (unless if a=0 (mod p)). $\endgroup$ – Douglas S. Stones Oct 14 '10 at 21:22
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HINT $\rm\ \ \ x^8 - 16\ =\ (x^2 - 2)\: (x^2 + 2)\: (x^4 + 4)\:.\ \:$ If the first two factors have no roots in $\rm\ \mathbb Z/p\ $ then $\:2, -2\:$ are nonsquares so their product $-4\: $ is a square, so $\rm\: i = \sqrt{-1} \in \mathbb Z/p\:$. Thus the third factor has a root since $\rm\ x^4 + 4\ $ has roots $\rm\: \pm 1\pm i\:$.

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  • $\begingroup$ I'm only learning elementary number theory, so I'm not too familiar with this. Could you please explain how $-4$ being a square implies $i=\sqrt{-1}\in\mathbb{Z}/p$? $\endgroup$ – yunone Oct 14 '10 at 23:27
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    $\begingroup$ If $j^2 \equiv -4$ then $(j/2)^2 \equiv -1$. Note: since p is odd, $1/2 \in \mathbb Z/p$, in fact $1/2 \equiv (p+1)/2$ $\endgroup$ – Bill Dubuque Oct 14 '10 at 23:31
  • $\begingroup$ The quartic factors even further: $x^8 - 16 = (x^2+2)(x^2-2)(x^2+2x+2)(x^2-2x+2)$. The last two factors both have discriminant $-4$, so if neither 2 nor $-2$ is a square mod $p$ then $-1$ is and thus the last two factors both have roots mod $p$. $\endgroup$ – KCd Oct 11 '11 at 2:35
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I usually set this as an exercise when teaching Number Theory. My hint is to ask the students: what are the solutions of $z^8=16$ in the complex numbers?

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Another solution, which doesn't need quadratic reciprocity but does use the elementary fact that the product of two quadratic nonresidues is a quadratic residue.

According to $p\bmod 8$,

  • if $p=1\mod 8$, $\pm 2$ are both quadratic residues
  • if $p=3\mod 8$, $-2$ is a quadratic residue but $2$ is not
  • if $p=5\mod 8$, $\pm 2$ are not quadratic residues
  • if $p=7\mod 8$, $2$ is a quadratic residue but $-2$ is not

If either $2$ or $-2$ is a quadratic residue, $4$ is a 4th power, so $16$ is an 8th power.

This accounts for all primes except those of the form $p=8k+5$, modulo which neither $2$ nor $-2$ is a quadratic residue. The product of two quadratic non-residues is a quadratic residue, so $-1=x^2$ for some $x$.

If $x=y^2$ for some $y$, $y^4=x^2=-1$ so $y^8=1$ so $y$'s order is 8. But the order of any element of a group divides the order of the group, which in this case is $\phi(p)=p-1=8k+4$, and $8\nmid 8k+4$.

So neither $2$ nor $x$ is a quadratic residue, so $2x$ is a quadratic residue $2x=y^2$ for some $y$, so $y^4=(2x)^2=4x^2=-4$ so $y^8=16$.

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For the sake of contradiction, suppose there exists a prime $p$ such that $x^8 \equiv 16 \pmod p$ has no solutions. We can factor $$x^8 - 16 = (x^2 + 2)(x^2 - 2)(x^2 + 2x + 2)(x^2 - 2x + 2).$$

Since this is not divisible by $p$, we must have $$\left ( \frac{2}{p} \right ) = \left ( \frac{-2}{p} \right ) = -1 \implies \left ( \frac{-1}{p} \right ) = 1.$$

But then $x^2 - 2x + 2 = (x-1)^2 + 1 \equiv 0 \pmod p$ has a solution, which is a contradiction.

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