Juliusz Brzezinski

On Sequences of Squares with Constant Second Differences

The aim of this paper is to study sequences of integers for which the second differences between their squares are constant. We show that there are infinitely many nontrivial monotone sextuples having this property and discuss some related problems.

On embedding numbers into quaternion orders.

A generalization of the chevalley-Hasse-Noether theorem from maximal orders to arbitrary Eichler orders in quaternion algebras is given. A stability property for the numbers of orbits for unit groups in quaternion orders acting on optimal embeddings of quadratic orders is proved. The results are applied to Siegels meanvalue of integral representations by genera of integral definite ternary quadratic forms.

On two classical theorems in the theory of orders

Abstract The paper contains generalizations of the Latimer-MacDuffee theorem and the Chevalley-Hasse-Noether theorem. It shows that the two theorems are closely related to each other by means of a duality, which depends on simultaneous actions of the idele groups on maximal orders in central simple algebras and on embeddings of maximal commutative subrings into such orders.

Examples and Selected Solutions

This chapters contains sample and in some cases complete solutions of some problems from Chaps. 1–15. We try to avoid solving standard problems formulated in these chapters, so that a solution is given as an example of how to handle similar problems to those presented in the text. Thus, the standard exercises in the text may be used as homework. Complete solutions to more special problems, as presented in this chapter, have at least three different functions. First, they contain a number of useful auxiliary results which are usually proved in the main texts of more standard textbooks. Second, some solutions are examples of how to work with the notions and approach similar problems (there is a rich selection of problems without solutions in Chap. 16). Third, some of the solutions presented in this chapter may be regarded as the last resort when serious attempts to solve a problem have been fruitless, or in order to compare one’s own solution to the one suggested in the book.

Proofs of the Theorems

This chapter contains the proofs to all theorems presented in the book. Only a few theorems, which are typically covered in an introductory course on groups, rings and fields are proved in the Appendix. A proof of the fundamental theorem of algebra is given in connection with the exercises in Chap. 13.

Hints and Answers

This chapter contains hints and answers to all exercises presented in Chaps. 1– 15 where an answer can be expected.

Solving Algebraic Equations

The aim of this chapter is to show how equations of degrees less than 5 can be solved. We highlight well-known formulae for the quadratic equation and show how to find similar formulae for cubic and quartic equations. We also explain why as early as the eighteenth century mathematicians started to doubt the possibility to find solutions for general quintic equations (or equations of higher degrees) using the four arithmetic operations and extracting roots applied to coefficients. We give examples of quantic equations for which such formulae exist (e.g. de Moivre’s quintics) and show that the ideas which work for equations of degrees up to 4 have no evident generalizations. We also briefly discuss “casus irreducibilis” related to cubic equations.

Automorphism Groups of Fields

In this chapter, we study automorphism groups of fields and introduce Galois groups of finite field extensions. The term “Galois group” is often reserved for automorphism groups of Galois field extensions, which we define and study in Chap. 9. The terminology used in this book is very common and has several advantages in textbooks (i.e. it is easier to formulate exercises). A central result of this chapter is Artin’s lemma, which is a key result in the modern presentation of Galois theory. In the exercises, we find Galois groups of many field extensions and we use also use this theorem for various problems on field extensions and their automorphism groups.

Solvability of Equations

In this chapter, we show that equations solvable by radicals are characterized by the solvability of their Galois groups. This immediately implies that general equations of degree 5 and above are not solvable by radicals. If one has a more modest goal to prove that the fifth degree general equation over a number field is not solvable by radicals, then there exists a simple argument by Nagell which only requires limited knowledge of field extensions and no knowledge of Galois theory. We consider Nagell’s proof in the exercises. This chapter further outlines Weber’s theorem on irreducible equations of prime degree (at least 5) with only two nonreal zeros, which are examples of non-solvable equations. We further discuss Galois’ classical theorem, which gives a characterization of irreducible solvable polynomials of prime degree. Both Galois’ and Weber’s results give examples of concrete unsolvable polynomials over the rational numbers. The solvability by real radicals in connection with “casus irreducibilis” is also discussed.

Polynomials and Irreducibility

In this chapter, we present facts on zeros of polynomials and discuss some basic methods to decide whether a polynomial is irreducible or reducible, including Gauss’ lemma, the reduction of polynomials modulo prime numbers ((irreducibility over finite fields), and Eisenstein’s criterion.