Extension of fields - Number of elements

Question

If K is an n-dimensional extension field of $Z_p$, what is the maximum possible number of elements in K?

I am a biginner and i dont understand all the procedure to proof that. I dont know how to start it.

Thanks for the help!!

Answer 1

$K$ is a $n$-dimensional vector space so it has $p^n$ elements. It is also the splitting field of $X^{p^n}-X$.

An element of $K$ is determined by its coordinates in a base $(e_1,...,e_n)$ of $K$ considered as a $Z_p$ vectors spaces, so you construct a map from $f:K\rightarrow (Z_p)^{n}$ defined by $f(x)=(x_1,...,x_n)$ where $x=\sum x_ie_i$ now show that this map is bijective since the coordinates of an element in a base are unique. The result follows from the fact that $(Z_p)^n$ has $p^n$ elements.

Let $K^*$ be the set of non zero elements of $K$, it is a finite group which has $p^n-1$ elements (it can be even shown that this group is cyclic) so you have for every $x\in K^*, x^{p^n-1}=1$ this implies that $xx^{p^n-1}=x$ or $x^{p^n}-x=0$ since $0^{p^n}=0$, you deduce that the polynomial $X^{p^n}-X$ has $p^n$ distinct roots in $K$, so $K$ is its splitting field.