For n at least 2, it is easy to show that the lower bound is greater than 1, so we obtain Minkowski's Theorem, that the discriminant of every number field, other than Q, is non-trivial.
The assertion that for n greater than or equal to 2, the lower bound is greater than 1 leads to Minkowski's Theorem, which concludes that all number fields, except the rational numbers Q, possess a non-trivial discriminant.