Fundamental Properties of the Galois Correspondence

Problem Statement: Let K is the splitting field of a polynomial f(x) over a field F and α_n be the roots of f in K. Let G be embedded as a subgroup of the symmetric group ς. We determined the Galois group G, and the subgroup. Approach: computed some auxiliary polynomials that had roots in K, where the permutation of a set was considered distinct. The Galois Theory was deduced using the primitive element and Splitting theorems. Results: The Galois extension K/L to identity L and its Galois group is a subgroup of G. which was referred to as the main theorem which we proved. Conclusion: Hence the findings suggest the need for computing more auxiliary polynomials that have roots.