The Research On The Problems Of Galois Group And Polynomials

In this paper, we firstly investigate the feature or the existence form of radical solutionsfor polynomial equations. When the degrees of the polynomials are quadratic, cubic, or quartic,we can get the solution by corresponding formula. Abel proved that for those indefinitecoefficients equations, if the degrees are not less than four, then the equations have no radicalsolutions. However, some higher degrees equations with definite coefficient have been foundhaving its radical solutions. Galois proved when the Galois group of the equation is solvablegroup, the equation has radical solutions. We can investigate the feature of radical solutionswhen the Galois group is Abel group, the super-solvable group or nilpotent group. We givesome supplement and explanation for the radical solutions of lower degrees equations via theidea of extensions in abstract algebra, and list these radical solutions. In addition, according tothe characteristics of the Galois groups and the idea of the extension, we find some methods tosolve some higher degrees equations. We mainly prove when the Galois group of the equationis Abel group, the super-solvable group and nilpotent group, its radical solution can be foundby cycle extension, and give the feature of radical solutions.Secondly, we get a method to criticize the irreducibility of polynomials via investigatingproperty of its Galois groups. Eisenstein criterion is useful in algebra to criticize whether somepolynomial, its coefficients are integral in the field of rational numbers, is reducible. Weinvestigate by the Galois groups of the polynomial equations to criticize the irreducibility ofthe polynomials, and prove if the Galois group of polynomial whose coefficient are integral,can be decomposed the direct product of its two subgroups, then it is provided withirreducibility. Besides, we verify that the inverse theorem is false. This theorem gives a newmethod to criticize the irreducibility of the polynomials.