O mistério por trás da resolução de equações polinomiais

Abstract

Solving polynomial equations was a problem that followed countless mathematicians over the centuries. As mathematics improved, formulas for Ąnding polynomial roots were developed. But the great mystery that plagued the mathematical community until the 19th century was that no one could Ąnd a solving formula for polynomial equations of degree 5 using only the operations of addition, subtraction, multiplication, division, exponentiation, together with the extraction of roots (known as solving by radicals). Several mathematicians tried to solve this problem, unsuccessfully, until a young mathematician named Evariste Galois (1811-1832) developed an innovative theory that solved this mystery. The objective of this work is to present the theory developed by Galois, showing why it is not possible to Ąnd a solving formula by radicals that provides the roots of polynomials of degree greater than or equal to 5. It is also intended to present several examples that demonstrate how this theory works. Finally, methods will be presented to calculate the Galois group of any polynomial of degree less than or equal to 5.