Galois theory, inverse problem of

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

If $k$ 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 $k$ 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

$$\frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}+\dotsb+\frac{x}{1!}+1=0$$

(partial sums of the expansion of the exponential function) have as Galois group the alternating group if $n\equiv0\pmod 4$, 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 $G$ as Galois group. As a solution one may select a field $K$ such that the discriminant of $K$ 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.

References

[1]	N.G. [N.G. Chebotarev] Tschebotaröw, "Grundzüge der Galois'schen Theorie" , Noordhoff (1950) (Translated from Russian)
[2a]	I.R. Shafarevich, "On the construction of a field with a given Galois group of order $l^\alpha$" Izv. Akad. Nauk SSSR , 18 : 2 (1954) pp. 261–296 (In Russian)
[2b]	I.R. Shafarevich, "The construction of an algebraic number field with a given solvable Galois group" Izv. Akad. Nauk SSSR , 18 : 3 (1954) pp. 525–578 (In Russian)

Comments

In recent years much progress has been made on the inverse problem of Galois theory over the base field $K=\mathbf Q$, see [a1]–[a3]. The technique is to realize first the group over the base field $\mathbf C(X)$, the field of rational functions in one variable over the complex numbers, which is well under control. Under certain conditions on the group one can replace $\mathbf C(X)$ by $\mathbf Q(X)$, using "descent" techniques. Finally one replaces $\mathbf Q(X)$ by $\mathbf Q$ using the Hilbert irreducibility theorem (cf. Hilbert theorem). In this way many finite simple groups have been realized as Galois groups over $\mathbf Q$ and over cyclotomic extensions of $\mathbf Q$.

References

[a1]	G.V. Belyi, "On extensions of the maximal cyclotomic field having a given classical Galois group" J. Reine Angew. Math. , 341 (1983) pp. 147–156
[a2]	B.H. Matzat, "Konstruktive Galoistheorie" , Lect. notes in math. , 1284 , Springer (1987)
[a3]	J.P.Serre, "Groupes de Galois sur $\mathbf Q$" Sem. Bourbaki , Exp. 689 (1987)