TITLE: Why do people study algebraic extension?
QUESTION [11 upvotes]: Yesterday, I learned Kronecker’s theorem and a finite extension. And now I’m studying the next chapter, Algebraic extension. 
I think the next theorem shows how important algebraic extension is 

Let $E$ be an extension field of a field $F$. Let $α$ be an element of $E$. If $α$ is algebraic over $F$, there exists a unique monic irreducible polynomial $p(x)$ in $F[x]$ such that $p(α)=0$ in $E$.

But I’m wondering there is another important reason why we should study algebraic extensions. 
Thanks for your help.

REPLY [3 votes]: To add to Joshua Ruiter's answer. 
You can teach a computer how to calculate with extensions field "exactly". If you see the $\sqrt{2}$ as an infinite decimal, your computer can only compute approximations (in finite time). But if you teach the computer that it is a pair $(0,1)$, with some algebraic properties, it will be able to give back exact answers to your calculations that involve $\sqrt{2}$.