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Abstract
Let ${\cal C}$ be any nonempty collection of finite extensions of Q, such that if $K = Q(\theta)\in {\cal C}$, then K is a normal extension of Q. Let $\Delta\sb{K}$ be the discriminant for K as an extension of Q. Define$$\eqalign{&\qquad\qquad\quad\quad d({\cal C}) = {\rm min}\{\vert\Delta\sb{K}\vert : K\in{\cal C}\}\cr&{\cal C}\sb{G} = \{K : K\ {\rm is\ a\ finite\ extension\ of}\ Q\ {\rm with}\ G(K,Q)\cong G\}}$$and$$d(G) = d({\cal C}\sb{G}).$$
One of the main objectives of our research is to find d(G) for all groups that satisfy $\vert G\vert\le 10$ and to identify for each case a field K such that $\vert\Delta\sb{K}\vert$ = d(G).
Our results will be derived based mainly on the known structure of the quadratic and cyclotomic fields, results derived for the biquadratic fields, tables of cubic and quintic fields of small discriminant, class field theory and character theory. Much research has been done on finding the minimum discriminant of algebraic number fields of given degree with a prescribed number of real and complex roots, but little has been done based on the group theoretical behavior of normal extensions.
