# Galois theory, inverse problem of

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The problem of constructing a finite normal extension of a given field with given Galois group (cf. Galois theory), and of stating the conditions which ensure the existence (and non-existence) of such an extension over .

If is the field of rational numbers, the problem becomes one of constructing a normal algebraic number field with given Galois group, and comes down to finding an algebraic equation over with the given Galois group. Such equations exist for the symmetric groups, and also for the alternating groups. I. Schur constructed equations for the alternating groups; it was shown, in particular, that equations of the form (partial sums of the expansion of the exponential function) have as Galois group the alternating group if , and the symmetric group as Galois group in other cases.

I.R. Shafarevich

used the arithmetical properties of algebraic number fields to show the existence of an extension of an algebraic number field with any solvable group as Galois group. As a solution one may select a field such that the discriminant of over the algebraic number field is relatively prime with any given integer, so that the number of solutions of the problem is infinite.

Considering the Galois groups of infinite extensions of a given field (cf. Galois topological group) makes it possible to solve the inverse problem of Galois theory in one stroke for special classes of fields: finite fields, local fields or fields of algebraic functions in one variable.

How to Cite This Entry:
Galois theory, inverse problem of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Galois_theory,_inverse_problem_of&oldid=11544
This article was adapted from an original article by S.P. Demushkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article