Why are algebraic numbers important and worth defining?

Solving equations is certainly a very natural and motivating problem in Mathematics. Historically, and practically, a most important class of equations are thise of the form $$ P(x)=0 $$ where $P(x)$ is a polynomial of any degree with coefficients rational numbers.

Two most important results of XIX century mathematics have been:

• there are some real numbers (e.g. $\pi$) that do not appear as solutions of equations of this sort;
• there's no explicit expression involving only the four basic operations and the radicals that give the solutions of these equations in general when the degree of $P(x)$ is $\geq5$.

Thus, the set $\bar{\Bbb{Q}}$ of algebraic numbers, namely the set of real (and complex, since Gauss) numbers that appear as solutions of equations of this sort is a proper subset of all real and complex numbers, but it is by no means explicit.

Thus, it becomes instantly an important, and very natural, object of mathematical inquiry.

One soon discovers that $\bar{\Bbb{Q}}$ is a field, i.e. it is closed under addition, multiplication and satisfies the usual properties of these operations, and moreover that it is algebraically closed, meaning that if we allowed to consider polynomial equations $P(x)=0$ where the coefficients of the polynomial are algebraic numbers, the set of solutions would be still $\bar{\Bbb{Q}}$.

Then, thanks to Galois, we see that the field $\bar{\Bbb{Q}}$ has lots of symmetries and that these symmetries allow--in principle--to classify all the intermediate subsets $${\Bbb{Q}}\subseteq K\subseteq\bar{\Bbb{Q}}$$ which are themselves fields.

The set $G_\bar{\Bbb{Q}}$ of these symmetries is actually a group and this group induces symmetries on many objects that arise from geometry, because it is very natural to consider algebraic subsets of the space (of any dimension) that are defined by polynomial equations, in as many variables as one wants, which have as coefficients rational, or algebraic, numbers.

This is just a quick hint on how $\bar{\Bbb{Q}}$ is a set of paramount importance in modern mathematics.