# Galois theory
---
Arose from the problem of solving polynomials of degree 5 by means of radicals.
In its modern formulation, it employs group theory to study polynomials made from some field.





\begin{definition}[Galois group]
\end{definition}




\begin{theorem}[Fundamental theorem of Galois theory]
\end{theorem}





\begin{theorem}
A polynomial is solvable by radicals iff its Galois group is solvable.
\end{theorem}


The famous Abel-Ruffini theorem can be proven as an application of this theorem. One first needs to calculate the Galois groups formed by quintic polynomials and then determine its solvability, during which one finds an unsolvable group!


\begin{theorem}[Abel-Ruffini theorem]
\end{theorem}