abel's theorem in problems and solutions - v.b. alekseev

[...]... Viète’s Theorem3 : The complex numbers and are the roots of the equation if and only if and Indeed, if and are roots of the equation then Eq (3) is satisfied, from which and Conversely, if and then, substituting and in the equation by their expressions in terms of and we obtain and therefore and are roots of the equation 2 The quadratic trinomial is a perfect square, i.e., for some complex number if and. .. brackets and collecting the terms of the same degree in we obtain the equation The coefficient of in this equation is equal to Therefore if we put after substituting we transform the equation into: where and are some polynomials in and Let be a root of Eq (6) Representing it in the form (where and are temporarily unknown) we obtain and We check whether it is possible to impose that In this case we obtain... and satisfy and By Viète’s theorem, for any such and (which may be complex) indeed exist, and they are the roots of the equation If we take such (still unknown) and then Eq (7) is transformed into Raising either terms of the equation to the third power, and comparing the obtained equation with Eq (8), we have 5 Introduction By Viète’s theorem and are the roots of the equation In this way where again... letter of its name in English (for example, = L) Is a mapping of the set of capitals onto the entire English alphabet? DEFINITION The mapping is called a one to one (or bijective) mapping of the set X into the set Y if for every in Y there exists a pre-image in X and this pre-image is unique 9 Consider the following mappings of the set of all integer numbers into the set of the non-negative integer numbers:... elements in an infinite cyclic group there corresponds the addition of integers (see 7) We come in this way to an important notion in the theory of groups: the notion of isomorphism 1.5 Isomorphisms DEFINITION Let two groups and be given with a bijective mapping 20 Chapter 1 from into (see §1.2) with the following property: if and in group then in group In other words, to the multiplication in there... elements of a finite group is called the order of the group Groups containing an infinite number of elements are called infinite groups Let us give some examples of infinite groups EXAMPLE 7 Consider the set of all integer numbers In this set we shall take as binary operation the usual addition We thus obtain a group Indeed, the role of the unit element is played by 0, because for every integer Moreover,... the holomorphic first integral and in many other insolvability problems of differential equations theory) I hope that the description of these ideas in the present translation of Alekseev s book will help the English reading audience to participate in the development of this new topological insolvability theory, started with the topological proof of the Abel Theorem and involving, say, the topologically... of the type: 2 in which , is called the generic algebraic equation of degree one variable For we obtain the linear equation in This equation has the unique solution for any value of the coefficients For we obtain the quadratic equation (in place of we write as learnt in school) Dividing both members of this equation by and putting and we obtain the reduced equation 2 For the time being the coefficients... again indicates one defined value of the square root Hence the roots of Eq (6) are expressed by the formula in which for each of the three values of the first cubic root4 one must take the corresponding value of the second, in such a way that condition be satisfied The obtained formula is named Cardano’s formula5 Substituting in this formula and by their expressions in terms of and subtracting we obtain... solve some problems, he must read their solutions in the Section Hints, Solutions, and Answers 2 We mean rotation of the plane around one axis perpendicular to the plane Groups 11 the triangle ABC into itself We put the transformation corresponding to in the intersection of the row corresponding to the transformation with the column corresponding to the transformation So, for example, in the selected . +08'00' ABEL’S THEOREM IN PROBLEMS AND SOLUTIONS This page intentionally left blank Abel’s Theorem in Problems and Solutions Based on the lectures of Professor V. I. Arnold by V. B. Alekseev Moscow. very grateful to V. I. Arnold for having made a series of important remarks during the editing of the manuscript. V. B. Alekseev Introduction We begin this book by examining the problem of solving. number of problems. The problems are posed directly within the text, so representing an essential part of it. The problems are labelled by increasing numbers in bold figures. Whenever the problem