# Existence and uniqueness of a subfield of an algebraic closure of a certain degree over a finite field

Currently, I am reading Rohrlich's article "Elliptic Curves and the Weil-Deligne Group" and got stuck in the beginning already.

Namely, let $k$ be a finite field of characteristic $p$ and cardinality $q$ and let $\bar{k}$ denote an algebraic closure of $k$. Then it is said that for an positive integer $n$ then there is a unique subfield $k_n$ of $\bar{k}$ of degree $n$ over $k$.

Could you explain me why it exists and why this is a unique subfield? Thank you.

• By basic results on finite fields the elements of $k_n$ are zeros of $x^{q^n}-x$. This polynomial has exactly $q^n$ zeros in $\overline{k}$, so $k_n$ gets specified as a subset. – Jyrki Lahtonen Jul 16 '18 at 14:33
• Do the answers/discussion e.g. here give enough details? Sorry if I sound a bit "less than enthusiastic". We have just covered this theme from so many angles so many times :-) – Jyrki Lahtonen Jul 16 '18 at 14:37