## Galois Theory

This chapter presents the most essential arguments of Galois in modern form, beginning with an examination of the extension field generated by all the roots of a given polynomial p(x) over a given field F. Classically, algebraists tried to solve real and complex polynomial equations by explicit formulas. But repeated attempts to obtain similar formulas which would solve general quintic (fifth-degree) equations proved fruitless. The reason for this was finally discovered by Evariste Galois, who showed that an equation is solvable by radicals if and only if the group of automorphisms associated with it is "solvable" in a purely group-theoretic sense. The automorphisms in question are those automorphisms of the extension field generated by all the roots of the equation, which leave fixed all the coefficients of the equation. By systematically using the properties of root fields, one can obtain a complete treatment of all fields with a finite number of elements (finite fields).