Typically, the algebraic closure of a non-algebraically closed field F is an infinite extension of F. However, this doesn't always have to happen: for example consider $\mathbb{R}$ inside $\mathbb{C}$. Are there any other examples? Yes: for example you can consider the index two subfield of the algebraic numbers, defined by intersecting with $\mathbb{R}$. However this is still similar to the first example: the degree of the extension is two, and we extract a square root of $-1$ to obtain the algebraic closure. The Artin-Schreier Theorem tells us that amazingly this is always the case: if $F$ is a field for which the algebraic closure is a non trivial finite extension $L$, then this forces F to have characteristic 0, L is degree two over $F$, and $L = F(i)$ for some $i$ with $i^2 = -1$. I.e. all such extensions "look like" $\mathbb{C} / \mathbb{R}$. In this expository talk we will give an overview of the proof of this theorem, and try to get some feeling for why this result is true.