There are many motivational problems related to the non-pure fields
extension corresponding to the algebraic numbers (1+(r)
DEGREES(1/n)) DEGREES(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 aut
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