This book is based on a course given by the author at Harvard University in the fall semester of 1988. The course focused on the inverse problem of Galois Theory: the construction of field extensions having a given finite group as Galois group. In the first part of the book, classical methods and results, such as the Scholz and Reichardt construction for p-groups, p != 2, as well as Hilbert's irreducibility theorem and the large sieve inequality, are presented. The second half is devoted to rationality and rigidity criteria and their application in realizing certain groups as Galois groups of regular extensions of Q(T). While proofs are not carried out in full detail, the book contains a number of examples, exercises, and open problems.

Topics in Galois Theory

Mathematics is full of dictionaries, one of the most well-thumbed being that between compact Riemann surfaces and complex algebraic curves; for example, branched coverings of surfaces and their groups of covering transformations can be translated into extensions of rational function fields and their Galois groups. An important problem is that of determining, for a given subfield K <= C, those surfaces defined over K, that is, given by a polynomial equation with coefficients in K. This is particularly interesting when K is the field Q of algebraic numbers, the algebraic closure in C of the rational field Q; being the union of all the algebraic number fields, Q has a particularly rich Galois theory, full of explicitly computable examples and tantalisingly difficult challenges, such as the Inverse Galois Problem, Hilbert's conjecture that every finite group is a Galois group over Q (see [35]). It follows from Weil's rigidity theorem [45] that a sufficient condition for a curve X to be defined over Q is that there should be a meromorphic function /? from X to the Riemann sphere (or projective line) £ = P*(Q = C U {oo} with at most three critical values (which we can take to be 0, 1 and oo). As a lemma for realising certain Chevalley groups as Galois groups over cyclotomic fields, Bely! [3] showed in 1979, by a simple but ingenious argument given in Section 2, that this condition is also necessary. Such a branched covering /?: Z ^ I , called a Belyi function, is completely determined by three permutations g0, g1 and g^ describing how the sheets are permuted by analytic continuation around 0, 1 and oo. They satisfy g0£i£oo = 1 a d generate a finite transitive permutation group, the monodromy group of /?, and conversely every such triple of permutations arises in this way. This has important consequences for uniformisation: for example, X is defined over Q if and only if X ^ %IK where °U is the upper half-plane and K is a subgroup of finite index in a triangle group A(/, m, n), or equivalently X = %/H where H has finite index in the principal congruence subgroup F(2) of the modular group (see Section 4). Since the time of Dyck [10] and Heffter [12,13] it has been known that maps (2-cell imbeddings of graphs) on compact orientable surfaces also correspond to such triples, satisfying the additional relation g\ = 1; the vertices, edges and faces correspond to the cycles of g0, gl and gx, with incidence given by non-empty intersection and orientation given by cyclic ordering. Thus maps correspond to subgroups of finite index in triangle groups A(/, 2, n); for regular maps, corresponding to normal subgroups, this is implicit in Chapter 8 of [8], and the general theory was developed independently in the 1970s by Malgoire and Voisin [24] and the authors [18,38].

Belyi Functions, Hypermaps and Galois Groups

In this paper, we consider two closely correlated subjects. One is a pro-i analogue of the braid group, and the other is a construction of the universal 1-adic power series for complex multiplications of Fermat type, or equivalently, for Jacobi sums. Both arise from, and constitute, a first step in the study of the canonical representation of the absolute Galois group GQ = Gal(Q/Q) in the outer automorphism group of the profinite fundamental group of PQ \{O,1,oo}.

Profinite braid groups, Galois representations and complex multiplications

We study the distribution of normalized spacings between the fractional parts of an^2, n=1,2,.... We conjecture that if a is \"badly approximable\" by rationals, then the sequence of fractional parts has Poisson spacings, and give a number of results towards this conjecture. We also present an example of a Diophantine number a for which the higher correlation functions of the sequence blow up.

The distribution of spacings between the fractional parts of n2α

We investigate the question of whether a query Q can be answered using a set V of views. We first define the problem in information-theoretic terms: we say that V determines Q if V provides enough information to uniquely determine the answer to Q. Next, we look at the problem of rewriting Q in terms of V using a specific language. Given a view language V and query language Q, we say that a rewriting language R is complete for Vto-Q rewritings if every Q e Q can be rewritten in terms of V e v using a query in R, whenever V determines Q. While query rewriting using views has been extensively investigated for some specific languages, the connection to the information-theoretic notion of determinacy, and the question of completeness of a rewriting language, have received little attention. In this paper we investigate systematically the notion of determinacy and its connection to rewriting. The results concern decidability of determinacy for various view and query languages, as well as the power required of complete rewriting languages. We consider languages ranging from first-order to conjunctive queries.

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Views and queries: determinacy and rewriting

This paper is devoted to the realization of certain types of Chevalley groups as the Galois group of extensions of certain cyclotomic fields. In addition, a criterion for an algebraic curve to be defined over an algebraic number field is given. Bibliography: 11 titles.

https://iopscience.iop.org/article/10.1070/IM1980v014n02ABEH001096/pdf

G V Belyĭ

Papers

On Galois Extensions of a Maximal Cyclotomic Field