Why do people study algebraic extension? 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.

I think the original motivation for studying field extensions was, as in the theorem you stated, to solve polynomials. One of the big results after a few lectures of algebraic field extensions is that every field can be embedded into a unique algebraically closed field, called the algebraic closure. 
Actually, solving equations is really the motivation for all the historical expansions of the concept "number." Think of it this way: we have the natural numbers, $1,2,3,\ldots$ and we can solve equations like $x+2 = 4$. But then we can pose equations like $x+ 2 = 2$ and $x+4 = 2$, so we want to extend our number system to include solutions to these, so we add zero and negative integers.
Then we notice we can pose equations like $2x = 4$, which has a nice solution $x=2$, and also $4x = 2$, which doesn't have a solution in the integers. So again, we extend our number system again to include things like $\frac 12$. Now we can solve any linear equation $ax+b = 0$ where $a,b \in \mathbb{N}$. 
We also have equations like $x^2 - 2 = 0$, and so we start adding irrational numbers like $\sqrt{2}$. We can complete the rational numbers to form the field of real numbers $\mathbb{R}$. But still, we have equations we can't solve, like $x^2 + 1 = 0$. To get a solution of this, we add the number $i = \sqrt{-1}$ and get the complex numbers. Going from $\mathbb{R}$ to $\mathbb{C}$ this way is an algebraic field extension.
As the theorem you stated says, this procedure is much more general than just $\mathbb{R}$ to $\mathbb{C}$: it says that given any abstract field $F$ and any polynomial equation with coefficients from that field, we can enlarge the "number system" that is $F$ to include a solution. 
EDIT: As was pointed out in a comment, the algebraic closure of a field is only unique up to a non-canonical isomorphism.
