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Is a given algebraic equation solvable by radicals? Can one solve a given algebraic equation of degree n using solutions of auxiliary algebraic equations of smaller degree and radicals? In this chapter, we discuss how Galois theory answers these questions (at least in principle).

The monodromy group of an algebraic function is isomorphic to the Galois group of the associated extension of the field of rational functions. Therefore, the monodromy group is responsible for the representability of an algebraic function by radicals. However, not only algebraic functions have a monodromy group. It is defined for the logarithm, arctangent, and many other functions for which the Galois group does not make sense. It is thus natural to try using the monodromy group for these functions instead of the Galois group to prove that they do not belong to a certain Liouville class. This particular approach is implemented in one-dimensional topological Galois theory [45–48, 50, 54, 56].

In topological Galois theory for functions of one variable (see Chap. 5), it is proved that the way the Riemann surface of a function is positioned over the complex line can obstruct the representability of that function by quadratures. This not only explains why many differential equations are not solvable by quadratures, but also gives the strongest known results on their unsolvability.

We give an account of the group logarithm and discuss a number of its applications. We begin by rehearsing some basic theory, and then go on to consider its use both in the theory of class groups of group rings and in the Galois structure of rings of algebraic integers. We conclude with a brief insight into its recent use in the construction of an adelic second Chern class for a non-commutative Riemann Roch theorem. The use of the group logarithm in non-commutative Iwasawa theory is discussed elsewherein this volume.

We introduce a notion of an idempotent semilinear space and consider two systems of linear-like equations. These systems are equivalent to systems of fuzzy relation equations with sup-*and \(\inf\)-→ compositions. We show that the theory of Galois connections can be successfully used in characterizing whether these systems are solvable and, if so, finding their solutions sets. Moreover, because the two types of systems of linear-like equations are dual according to this theory, it is sufficient to investigate only one system.

We introduce a notion of an idempotent semilinear space and consider two systems of linear-like equations. These systems are equivalent to systems of fuzzy relation equations with sup-* and inf-→ compositions. Moreover, because the two types of systems of linear-like equations are dual according to this theory, it is sufficient to investigate only one system.

A bridge between Lie symmetry groups for differential equations and Galois groups for algebraic equations is suggested. It is based on calculation of Lie symmetries for algebraic equations and their restriction of the roots of the equations under consideration. The approach is illustrated by several examples. An alternative representation of Lie symmetries, called the Galois representation, is provided for differential equations.

Our purpose is to prove that certain concepts of mathematical physics involving partial differential equations (PDE) can only be clearly understood in the framework of a new branch of mathematics called “differential algebraic geometry”.By this way, we give a new setting for the inverse Backlund problem and relate it to the differential Galois theory.

We give an overview of basic properties of Galois connections between sets of relations and sets of functions or generalized functions. First we focus on the Galois connections lnv --- Pol and lnv --- mPol . Then we use these results to provide some tools for the representation of several closure operators on relations as closure operators of some Galois connections.

We give here a comprehensive treatment of the mathematical theory of perturbative renormalization (in the minimal subtraction scheme with dimensional regularization), in the framework of the Riemann–pHilbert correspondence and motivic Galois theory. We give a detailed overview of the work of Connes–Kreimer [31], [32]. We also cover some background material on affine group schemes, Tannakian categories, the Riemann–Hilbert problem in the regular singular and irregular case, and a brief introduction to motives and motivic Galois theory. We then give a complete account of our results on renormalization and motivic Galois theory announced in [35].

We describe methods for explicit computation of Galois groups of certain tamely ramified p-extensions. In the finite case this yields a short list of candidates for the Galois group. In the infinite case it produces a family or few families of likely candidates.

We generalize a recent improvement for the bounds of Weil sums over Galois rings of characteristic p2 to Galois rings of any characteristic pl. Our generalization is not as strong as for the case p2 and we indicate the reason. We give a class of homogeneous weights, including the homogeneous weight defined by Constantinescu and Heise, and we show their relations. We also give an application of our improvements on the homogeneous weights of some codewords.

In this chapter we give the basic algebraic results from the differential Galois theory of linear differential equations. Other presentations of some or all of this material can be found in the classics of Kaplansky [151] and Kolchin [162] (and Kolchin’s original papers that have been collected in [25]) as well as the recent book of Magid [183] and the papers [231] and [173].

Linear differential equations form the central topic of this volume, Galois theory being the unifying theme. A large number of aspects are presented: algebraic theory especially differential Galois theory, formal theory, classification, algorithms to decide solvability in finite terms, monodromy and Hilbert's 21st problem, asymptotics and summability, the inverse problem and linear differential equations in positive characteristic. The appendices aim to help the reader with concepts used, from algebraic geometry, linear algebraic groups, sheaves, and tannakian categories that are used. This volume will become a standard reference for all mathematicians in this area of mathematics, including graduate students.

We will first sketch the contents of this chapter. Let (δ−A be a matrix differential equation over C({z}). Then there is a unique (up to isomorphism over C({z})) quasi-split equation δ−B that is isomorphic, over C((z)), to δ−A (cf. Proposition 3.41). This means that there is a \( \hat{F} \notin G{L_n}\left( {C\left( {(z)} \right)} \right) \) such that \( {\hat{F}^{{ - 1}}}\left( {\delta - A} \right)\hat{F} = \delta - B \). In the following, δ−A, δ−B, and \(\hat{F}\) are fixed and the eigenvalues of δ−A, δ−B are denoted by q1,…, q s .

Assuming the generalized Riemann hypothesis, we prove the following complexity bounds: The order of the Galois group of an arbitrary polynomial f(x) ∈ ℤ[x] can be computed in P# P. Furthermore, the order can be approximated by a randomized polynomial-time algorithm with access to an NP oracle. For polynomials f with solvable Galois group we show that the order can be computed exactly by a randomized polynomial-time algorithm with access to an NP oracle. For all polynomials f with abelian Galois group we show that a generator set for the Galois group can be computed in randomized polynomial time.

Fixed point calculus is about the solution of recursive equations defined by a monotonic endofunction on a partially ordered set. This tutorial presents the basic theory of fixed point calculus together with a number of applications of direct relevance to the construction of computer programs. The tutorial also summarises the theory and application of Galois connections between partially ordered sets. In particular, the intimate relation between Galois connections and fixed point equations is amply demonstrated.

Algebraic properties of logical relations on partially ordered sets are studied. It is shown how to construct a logical relation that extends a collection of base Galois connections to a Galois connection of arbitrary higher-order type. "Theorems-for-free" is used to show that the construction ensures safe abstract interpretation of parametrically polymorphic functions.

A Galois theory of a category R is a contravariant functor of R into a “simpler” category R′, where certain properties of the objects and morphisms of R are reflected in R′.

Galois theory stands at the cradle of modern algebra and interacts with many areas of mathematics. The problem of determining Galois groups therefore is of interest not only from the point of view of number theory (for example see the article [39] in this volume), but leads to many questions in other areas of mathematics. An example is its application in computer algebra when simplifying radical expressions [32].

We show that the usual constructions of bent functions, when they are suitably modified, allow constructions of correlation-immune and resilient functions over Galois fields and, in some cases, over Galois rings.

Patterns of the distribution of elements in words of linear codes over a Galois ring and in their representations over a Galois field are investigated. Often they may be evaluated using numbers of some special solutions of the equation defined by the trace-function on a Galois ring. Here the solutions of such an equation over a Galois ring R=GR(q2,4) of characteristic 4 are enumerated. It allows us in particular to describe the complete weight enumerators of the base linear code K R (m) and the appropriate Kerdock code K q (m+1) over a Galois field of q=21 elements.Results based on properties of special quadrics over GF(21) arise by description of the 2-adic decomposition of the trace-function.

This paper studies rational and Liouvillian first integrals of homogeneous linear differential systems Y′=AY over a differential field k. Following [26], our strategy to compute them is to compute the Darboux polynomials associated with the system. We show how to explicitly interpret the coefficients of the Darboux polynomials as functions on the solutions of the system; this provides a correspondence between Darboux polynomials and semi-invariants of the differential Galois groups, which in turn gives indications regarding the possible degrees for Darboux polynomials (particularly in the completely reducible cases). The algorithm is implemented and we give some examples of computations.

We propound a systematic and formal method to compute the Galois group of a non-necessarily irreducible polynomial: we proceed by successive inclusions, using mostly computations on scalars (and very few on polynomials). It is based on a formal method of specialization of relative resolvents: it consists in expressing the generic coefficients of the resolvent using the powers of a primitive element, thanks to a quadratic space structure; this reduces the problem to that of specializing a primitive element, which we are able to do in the case of the descending by successive inclusions. We incidentally supply a way to make separable a resolvent.

Let k be a number field which contains the fifth roots of unity and K/k an A5-extension. According to Klein and Serre K/k can be described by adjunction of the 5-torsion points of a suitable elliptic curve E. It is well known that there exists such a curve E defined over k if and only if there exists a quadratic overfield M of K such that M is Galois over k with group

On the experimental verification of the artin conjecture for 2-dimensional odd galois representations over Q liftings of 2-dimensional projective galois representations over Q.- A table of A5-fields.- A. Geometrical construction of 2-dimensional galois representations of A5-type. B. On the realisation of the groups PSL2(1) as galois groups over number fields by means of l-torsion points of elliptic curves.- Universal Fourier expansions of modular forms.- The hecke operators on the cusp forms of ?0(N) with nebentype.- Examples of 2-dimensional, odd galois representations of A5-type over ? satisfying the Artin conjecture.

In this paper we compute the differential Galois group of a third order linear differential equation whose existence was predicted by F. Klein [9] and whose construction is due to A. Hurwitz [7]. The aim of this paper is to apply the results of [15] in order to prove, starting only with the equation, that the simple group of 168 elements is the the differential Galois group of this equation.

The goal of these lectures is to explain the fundamentals of Galois theory and algebraic number theory and to get the reader to the point that he or she can make routine calculations. The following material has little in the way of prerequisites. Many examples will be given along the way and sometimes the examples will serve instead of formal proofs. In the last Section, we introduce the zeta functions of algebraic number fields. These functions can be factored into products of L-functions according to representations of Galois groups. Especially here, we will proceed by example. These lectures have been abstracted from my long promised forthcoming book [2]. A complete treatment of any of the topics here is somewhat beyond the length requirements of these lectures although we will come surprisingly close in some instances. I would also recommend Hecke’s book [1] which has been a classic for almost 70 years.

Perhaps the easiest description of differential Galois theory is that it is about algebraic dependence relations between solutions of linear differential equations. To clarify this statement, let us consider three examples. First consider the differential equation $$z(1-z){y}''+(\frac{1}{2}-\frac{7}{6}z){y}'+\frac{11}{3600}y = 0$$ (1.1)

We give a new version of Galois theory in categories in which normal extensions are replaced by arbitrary extensions for which the “pullback functor” is monadic, and their Galois groupoids are replaced by internal pregroupoids; we obtain the “fundamental theorem of Galois theory” using just simple remarks on internal precategories and change of universe for internal functors.

These notes arose out of an attempt to explain to myself a relatively complicated construction, that of the @tale topology of a scheme, in terms with which I was more familiar. These terms may not be so familiar to the reader, who might find these notes rather pointless in consequence, but I hope that I can put across some arguments why I think my way of looking at these constructions is a natural one.

The main purpose of this chapter is the proof of Theorems 5 and 7 and of Proposition I.6.2. We shall proceed essentially by reduction to local fields. On the way we shall then have to derive results, on resolvents and Galois Gauss sums etc., which are of independent interest.

The usual Galois correspondence between subgroups of Galois groups of finite Galois extensions and intermediate fields is not valid for infinite Galois extensions. The Krull topology restores this correspondence for closed subgroups (Proposition 1.8).