# Solution to equation in field

#### featherbox

Find all solutions to the equation $$\displaystyle x^{16} = 1$$ in the field $$\displaystyle Z_{17}$$.

A solution or a starting point would be greatly appreciated.

#### xixi

Find all solutions to the equation $$\displaystyle x^{16} = 1$$ in the field $$\displaystyle Z_{17}$$.

A solution or a starting point would be greatly appreciated.
Since $$\displaystyle \mathbb{Z}_{17}$$-{0} is a finite group of order 16 , so for every nonzero $$\displaystyle x\in \mathbb{Z}_{17}$$ we have $$\displaystyle x^{16} = 1$$ , so the solution is $$\displaystyle \mathbb{Z}_{17}$$-{0} .

#### featherbox

I got that because $$\displaystyle 16=-1$$ in the field, we have the equation now being $$\displaystyle x^{-1}=1$$, and seeing as all elements in the field have multiplicative inverses all elements within $$\displaystyle Z_{17}$$ are solutions.

#### xixi

I got that because $$\displaystyle 16=-1$$ in the field, we have the equation now being $$\displaystyle x^{-1}=1$$, and seeing as all elements in the field have multiplicative inverses all elements within $$\displaystyle Z_{17}$$ are solutions.
you can't say $$\displaystyle x^{16}=x^{-1}$$ , as I said $$\displaystyle x^{16}=1$$ and from $$\displaystyle x^{-1}=1$$ you can just conclude that $$\displaystyle x=1$$ and nothing else !

#### featherbox

you can't say $$\displaystyle x^{16}=x^{-1}$$ , as I said $$\displaystyle x^{16}=1$$ and from $$\displaystyle x^{-1}=1$$ you can just conclude that $$\displaystyle x=1$$ and nothing else !
Thats not what I said at all.
The solutions would be 1,2,...16. NOT just 1.
This is because $$\displaystyle x^{-1}=1$$ for $$\displaystyle x=1,2...,16$$

Why can't $$\displaystyle x^{16}=x^{-1}$$?

#### xixi

Thats not what I said at all.
The solutions would be 1,2,...16. NOT just 1.
This is because $$\displaystyle x^{-1}=1$$ for $$\displaystyle x=1,2...,16$$

Why can't $$\displaystyle x^{16}=x^{-1}$$?
from $$\displaystyle x^{-1}=1$$ we have $$\displaystyle x=1$$ because the only element whose inverse is 1 is itself 1 . Moreover since the order of each element divides the order of the group and the order of this group is 16 then $$\displaystyle x^{16}=1$$ and $$\displaystyle x^{16}=x^{17}x^{-1}$$ but $$\displaystyle x^{17}=x$$ and so $$\displaystyle x^{16}=x.x^{-1}=1$$ and as you see it won't be $$\displaystyle x^{-1}$$.