I found this definition in a book and I did not understand the meaning of it There exists a field of order q if and only if q is a prime power (i.e., $q = p^r$]) with p prime and r ∈ N. Moreover, if q is a prime power, then there is, up to relabelling, only one field of that order.
what mean r here, what is value for this variable. I know Any prime number only divisible by itself and one. How it take power.
On
Given a prime $p,$ the set often written $0,1,\ldots, p-1$ is a field under ordinary addition and multiplication modulo $p$ is a field. Call it $F.$ Given a polynomial $f(x)$ with integer coefficients and degree $r,$ such that $f(x)$ is irreducible in $F,$ (and the degree $r$ coefficient is not divisible by $p$) the thing we write as $F[x]/f(x),$ makes another field, this time with $p^r$ elements. The standard basis of this over $F,$ as a vector space, is $$ \{x^{r-1}, x^{r-2}, \ldots, x^2, x,1. \} $$
Oh, well, there is more to it. An element in this field can be taken to be a polynomial of degree at most $r-1,$ with coefficients in $0,1,\ldots, p-1.$ Addition is still just modulo $p,$ but multiplication needs to include the reduction given by subtracting off polynomial multiples of $f(x),$ which are considered to be zero in this setting.
I will focus on the meaning of "prime power":
To say that $q$ is a prime power simply means that there exists a positive integer $r$ and a prime number $p$ such that $q=p^r$. It is not claiming that $q$ is a prime, only that it is only divisible by a single prime.
For example, every prime number $p$ is a prime power by taking $r=1$, as $p=p^1$. $2,4,8,16,32$ are all prime powers, because they are equal to $2^1,2^2,2^3,2^4,2^5$, respectively.
Non-examples include $6$ because there is no prime $p$ and positive integer $r$ such that $6=p^r$ (this follows from the Fundamental Theorem of Arithmetic).