Hilbert's irreducibility theorem then implies that an infinite set of rational numbers give specializations of whose Galois groups are over the rational field.
2.
It is an open problem to prove the existence of a field extension of the rational field "'Q "'with a given finite group as Galois group.
3.
The characterization is the following : a complex number is constructible if and only if it lies in a field at the top of a finite tower of quadratic extensions, starting with the rational field.
4.
If the Lie group G is obtained as the real points of an algebraic group \ mathbf G over the rational field \ mathbb Q then the \ mathbb Q-rank of \ mathbf G has also a geometric significance.
5.
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