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For any finite field $F_q$ where $q$ is a prime power, I want to calulate the number of powers. Specifically, for 3$\le k \le q-1$, how to determine the size of the set: $\{x^k \mid x \in F_q\}$.

Since the order of $F_q^{\times}$ is $q-1$, I only need to consider the case 2$ \le k \le q-1$. Also I have known the case $k=2$.

Thanks for any advise.

addition:I read the post:Number of elements which are cubes/higher powers in a finite field. . Maybe this problem is to determine the kernel of the group endomorphism: $$\phi:F_q^\times \rightarrow F_q^\times \text{ sending } x\mapsto x^k$$

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Zongxiang Yi
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  • You might get some insight from the case $k=2$, which gives the set of so-called quadratic residues. – hardmath Nov 12 '16 at 13:10
  • for $k=2$, I have $x^2-1=(x+1)(x-1)$. But it's hard for $x^k-1$. And I don't know how many roots of $x^k-1=0$ lies in $F_q$. Any hints? – Zongxiang Yi Nov 12 '16 at 13:24
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    The group $F_q^\times$ is cyclic of order $q-1$. What do you know about the number of powers in a cyclic group? Do review the theory of subgroups of finite cyclic groups. The answer is hidden in there. – Jyrki Lahtonen Nov 12 '16 at 13:31
  • @hardmath thanks. Maybe there are some connections with higher reciprocity law over finite field. – Zongxiang Yi Nov 12 '16 at 14:01
  • @JyrkiLahtonen I intuit that it's about the structure of finite field$(F_q,+,\times)$ but not only about that of $F_q^\times=(F_q,\times)$. Expect for your details. Thanks anyway. – Zongxiang Yi Nov 12 '16 at 14:12
  • @JyrkiLahtonen Given $x$, to caculate $k$ such that $x^k=1$ is a problem. Given $k$, to caculate $x$ such that $x^k=1$ is another problem. It do seem that the two problem have some connections. Should I sum the number of elements whose order if a factor of $k$? Since one element has only one order, $D_i\bigcap D_j = \emptyset$ where $D_i$ denote the set of elements whose order is $i$. That's when $\gcd(k,q-1)=1$, $|{x^k \mid x\in F_q}|$. Am I right? – Zongxiang Yi Nov 12 '16 at 14:36
  • The linked question does generalize. If you study cyclic groups you will learn that raising to power $k$ in a cyclic group of order $n$ has kernel the size of $\gcd(k,n)$. With questions like this I leave it to the asker to connect a few dots, so no detailed answer from me coming. You are well on your way there though. Good job finding that other question! (+1) – Jyrki Lahtonen Nov 12 '16 at 14:37
  • And, if you find the order of the kernel, then you can apply the homomorphism theorem to find the order of the image. – Jyrki Lahtonen Nov 12 '16 at 14:39
  • I see. Thanks a lot. – Zongxiang Yi Nov 12 '16 at 14:42

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