Applications of Galois Theory

This paper gives an insight to the Galois theory and discusses its applications in both pure and applied mathematics. First, the Fundamental theorem of Galois theory is applied to compute the Galois groups of polynomials and to prove the non-existence of a formula for solving a polynomial equation in rational coefficients having degree n ≥ 5. Then the Galois fields which are finite fields are applied to the error-correcting codes and cryptography in computer science. There are no general rules to compute the Galois groups of polynomials of degree more than four. Two new examples of Galois groups of polynomials of degree greater than four are introduced and the concept of Galois group of a single variable polynomial is extended to the Galois group of a multi-variable polynomial.