This chapter deals with Galois groups and automorphism of number fields and several kinds of function fields. It also deals with computing the subfields of these kinds of fields. While these problems are closely related from a theoretical point of view (basically, everything is determined by the Galois group), as algorithmic problems they are quite different.

The first task, that of computing automorphisms of normal extensions of Q (and of abelian extensions of number fields) can be thought of a special case of factorisation of polynomials over number fields: the automorphisms of a number field are in one-to-one correspondence with the roots of the defining equation in the field. However, the computation follows a different approach and is based on some combinatorial properties. It should be noted, though, that the algorithms only apply to normal fields; i.e., they cannot be used to find non-trivial automorphisms of non-normal fields!

The second task, namely that of computing the Galois group of the normal closure of a number field, is of course closely related to the problem of computing the Galois group of a polynomial. The method implemented in Magma allows the computation of Galois groups of polynomials (and number fields) of arbitrarily high degrees and is independent of the classification of transitive permutation groups. The result of the computation of a Galois group will be a permutation group acting on the roots of the (defining) polynomial, where the roots (or approximations of them) are explicitly computed in some suitable p-adic field; thus the splitting field is not (directly) part of the computation. The explicit action on the roots allows one, for example, to compute algebraic representations of arbitrary subfields of the splitting field, even the splitting field itself, provided the degree is not too large.

The last main task dealt with in this chapter is the computation of subfields of a number field. While of course this can be done using the main theorem of Galois theory (the correspondence between subgroups and subfields), the computation is completely independent; in fact, the computation of subfields is usually the first step in the computation of the Galois group. The algorithm used here is mainly combinatorical.

Finally, this chapter also deals with applications of the Galois theory:

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the computation of subfields and subfield towers of the splitting field
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solvability by radicals: if the Galois group of a polynomial is solvable, the roots of the polynomial can be represented by (iterated) radicals.
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basic Galois-cohomology; i.e., the action of the automorphisms on the ideal class group, the multiplicative group of the field and derived objects.
V2.28, 13 July 2023