Relative Quadratic Extension over a Pure Cubic Field: This Article Will Help You to Determine the Integral Basis of the Extended Field over the Rational Field Q

There are many motivational problems related to the non-pure fields extension corresponding to the algebraic numbers (1+(r)^(1/n))^(1/m), where m and n are positive integers. Here we take the extended field K over the field of rational numbers Q of degree n correspond to the inner nth root of the algebraic number and then the relative extension of degree m is taken over field K. If we interchange these nth and mth root then the whole structure and the resulting Hasse diagram change completely. In chapter 4 We have posed an open problem for the non-pure sextic field whose Galois closure is of extension degree 36. Since there are 14 groups of order 36 out of which four are abelian and ten are non-abelian and our group of automorphism is non-abelian so it is one of the ten. We had not only found this group but also create the correspondence between the Hasse diagram of subfields of Galois closure and the subgroups of group of automorphism.