Before you read: What names, to your knowledge, are associated with the theory of numbers? Do you know what their contributions were?

I. Look through the article quickly. What names are mentioned to have worked on the theory of numbers? Does this list of names coincide with yours?

While the theory of elliptic functions typifies the 19th century's enthusiasm for pure mathematics, some contemporary mathematicians said that the simultaneous developments in number theory carried that enthusiasm to excess. Nonetheless, during the 19th century the algebraic theory of numbers grew from being a minority interest to its present central importance in pure mathematics. The earlier investigations of Fermat had eventually drawn the attention of Euler and Lagrange. Euler proved some of Fermat's unproved claims and discovered many new and surprising facts; Lagrange not only supplied proofs of many remarks that Euler had merely conjectured but also worked them into something like a coherent theory. For example, it was known to Fermat that the numbers which can be written as the sum of two squares are either the number 2, squares themselves, primes of the form 4n + 1, or else products of these numbers. Thus 29, which is 4 × 7 + 1, is 5^{2} + 2^{2}, but 35, which is not of this form, cannot be written as the sum of two squares. Euler had proved this result and had gone on to consider similar cases, such as primes of the form x^{2} + 2y^{2}, or x^{2} + 3y^{2}. But it was left to Lagrange to provide a general theory covering all expressions of the form ax^{2} + bxy + cy^{2}, quadratic forms, as they are called.

Lagrange's theory of quadratic forms had made considerable use of the idea that a given quadratic form could often be simplified to another with the same properties but with smaller coefficients. To do this in practice, it was often necessary to consider whether a given integer left a remainder that was a square when it was divided by another given integer. (For example, 48 leaves a remainder of 4 upon division by 11, and 4 is a square.) Legendre discovered a remarkable connection between the question, “Does the integer p leave a square remainder on division by q?” and the seemingly unrelated question, “Does the integer q leave a square remainder upon division by p?” He saw, in fact, that, when p and q are primes, both questions have the same answer unless both primes are of the form 4n - 1. Because this observation connects two questions in which the integers p and q play mutually opposite roles, it became known as the law of quadratic reciprocity. Legendre also gave an effective way of extending his law to cases when p and q are not primes.

All this work set the scene for the emergence of Gauss, whose Disquisitiones Arithmeticae not only consummated what had gone before but also directed number theorists in new and deeper directions. He rightly showed that Legendre's proof of the law of quadratic reciprocity was fundamentally flawed and gave the first rigorous proof. His work suggested that there were profound connections between the original question and other branches of number theory, a fact that he perceived to be of signal importance for the subject. He extended Lagrange's theory of quadratic forms by showing how two quadratic forms can be “multiplied” to obtain a third. Later mathematicians were to rework this into an important example of the theory of finite commutative groups. And in the long final section of his book Gauss gave the theory that lay behind his first discovery as a mathematician: that a regular 17-sided figure can be constructed by circle and straightedge alone.

The discovery that the regular “17-gon” is so constructible was the first such discovery since the Greeks, who had known only of the equilateral triangle, the square, the regular pentagon, the regular 15-sided figure, and the figures that can be obtained from these by successively bisecting all the sides. But what was of much greater significance than the discovery was the theory that underpinned it, the theory of what are now called algebraic numbers. It may be thought of as an analysis of how complicated a number may be while yet being amenable to an exact treatment.

The simplest numbers to understand and use are the integers and the rational numbers; the irrational numbers seem to pose problems. Famous among these is √2. It cannot be written as a finite or repeating decimal (because it is not rational), but it can be manipulated algebraically very easily. It is only necessary to replace every occurrence of (√2)^{2} by 2. In this way expressions of the form m + n√2, where m and n are integers, can be handled arithmetically. These expressions have many properties akin to those of whole numbers, and one can even define prime numbers of this form; therefore they are called algebraic integers. In this case they are obtained by grafting onto the rational numbers a solution of the polynomial equation x^{2} - 2 = 0. In general an algebraic integer is any solution, real or complex, of a polynomial equation with integer coefficients in which the coefficient of the highest power of the unknown is 1.

Gauss's theory of algebraic integers led to the question of determining when a polynomial of degree n with integer coefficients can be solved given the solvability of polynomial equations of lower degree but with coefficients that are algebraic integers. For example, he regarded the coordinates of the 17 vertices of a regular 17-sided figure as complex numbers satisfying the equation x^{17} - 1= 0, and thus as algebraic integers. One such integer is z = 1. He showed that the rest are obtained by solving a succession of four quadratic equations. Because solving a quadratic equation is equivalent to performing a construction with a ruler and compass, as Descartes had shown long before, Gauss had shown how to construct the regular 17-gon.

Inspired by Gauss's works on the theory of numbers, a growing school of mathematicians was drawn to the subject. Like Gauss, the German mathematician Ernst Eduard Kummer sought to generalize the law of quadratic reciprocity to deal with questions about third, fourth, and higher powers of numbers. He found that his work led him in an unexpected direction, toward a partial resolution of Fermat's last theorem. In 1637 Fermat wrote in the margin of his copy of Diophantus' Arithmetica the claim to have a proof that there are no solutions in positive integers to the equation x^{n} + y^{n} = z^{n} if n > 2. However, no proof was ever discovered among his notebooks.

Kummer's approach was to develop the theory of algebraic integers. If it could be shown that the equation had no solution in suitable algebraic integers, then a fortiori there could be no solution in ordinary integers. He was eventually able to establish the truth of Fermat's last theorem for a large class of prime exponents n (those satisfying some technical conditions needed to make the proof work). This was the first significant breakthrough in the study of the theorem. Together with the earlier work of the French mathematician Sophie Germain, it has enabled mathematicians to establish Fermat's last theorem for every value of n from 3 to 4,000,000. However, Kummer's way around the difficulties he encountered further propelled the theory of algebraic integers into the realm of abstraction. It amounted to the suggestion that there should be yet other types of integers, but many found these ideas obscure.

In Germany Richard Dedekind patiently created a new approach, in which each new number (called ideal) was defined by means of a suitable set of algebraic integers in such a way that it was the common divisor of the set of algebraic integers used to define it. Dedekind's work was slow to gain approval, yet it illustrates several of the most profound features of modern mathematics. It was clear to Dedekind that the ideal algebraic integers were the work of the human mind. Their existence can neither be based on nor deduced from the existence of physical objects, analogies with natural processes, or some process of abstraction from more familiar things. A second feature of Dedekind's work was its reliance on the idea of sets of objects, such as sets of numbers, even sets of sets. Dedekind's work showed how basic the naive conception of a set could be. The third crucial feature of his work was its emphasis on the structural aspects of algebra. The presentation of number theory as a theory about objects that can be manipulated (in this case, added and multiplied) according to certain rules akin to those governing ordinary numbers was to be a paradigm of the more formal theories of the 20th century.

II. What are the meanings of italicized words? Look them up in the English-Russian dictionary and provide the best translation of the sentences:

1.…the algebraic theory of numbers grew from being a minority interest to its present central importance in pure mathematics.

2.…a growing school of mathematicians was drawn to the subject.

3. But it was left to Lagrange to provide a general theory covering all expressions…

4. Legendre discovered a remarkable connection between the question…

III. Analyse the functions of the Infinitives and translate the sentences into Russian:

1. Euler had proved this result and had gone on to consider similar cases…

2. Ernst Eduard Kummer sought to generalize the law of quadratic reciprocity to deal with questions about third, fourth, and higher powers of numbers.

3. Dedekind's work was slow to gain approval…

IV. Read the article and answer the questions:

1. Were mathematicians enthusiastic about the theory of numbers in the 19^{th} century?

2. How did it develop in the 19^{th} century?

3. How did Euler and Lagrange contribute to it?

4. What did Lagrange's investigations finally result in?

5. What did he extend his law to?

6. What did Gauss show?

7. How did he extend Lagrange's theory?

8. What were mathematicians to rework this into later?

9. How did Gauss come to the theory of algebraic numbers?

10. What did Gauss's theory of algebraic integers lead to?

11. What did Ernst Eduard Kummer seek to do?

12. What was he eventually able to establish the truth of? How important was it?

13. Where did Kummer propel the theory of algebraic integers?

14. What approach did Richard Dedekind create?

15. What was clear to him about ideal numbers?

16. What did Dedekind's work show?

17. What was the third crucial feature of his work?

18. How did it change mathematics in the 20^{th} century?

V. Read the article again and single out the key information in each paragraph.

VI. Speak about the development of the theory of numbers using the flowchart:

Fermat Euler Lagrange Gauss Kummer Dedekind

VII. Write out and learn all the possible combinations with the word "equation".

VIII. Fill in the missing forms:

Noun

Verb

Adjective

Adverb

consider

reciprocity

rigorous

observe

successively

generalize

constructible

IX. Write out all the possible words and phrases pertaining to the thematic group 'number'

X. Add all the possible attributes to the spidergram with the noun "integer":

integer

XI. Match the words with their synonyms:

resolution quantity

supply look at

consider include

cover easy

simple provide

number solution

XII. Suggest the Russian counterparts for the following words and word combinations:

pure mathematics, algebraic theory of numbers, to draw attention, to supply a proof of, a coherent theory, theory of quadratic forms, a square remainder, the law of quadratic reciprocity, to set the scene for, to be of signal importance, the theory of finite commutative groups, an equilateral triangle, a regular pentagon, the theory of algebraic numbers, to pose a problem, a repeating decimal, an algebraic integer, a polynomial equation, a complex number, a partial resolution of, a fortiori, to establish the truth, to amount to, to create a new approach, the common divisor, to gain approval, the work of the human mind, ideal number, to be based on, reliance on, a set of sets, a crucial feature, emphasis on.

Another subject that was transformed in the 19th century was the theory of equations. Ever since Tartaglia and Ferrari in the 16th century had found rules giving the solutions of cubic and quartic equations in terms of the coefficients of the equations, formulas had unsuccessfully been sought for equations of the fifth and higher degrees. At stake was the existence of a formula that expresses the roots of a quintic equation in terms of the coefficients. This formula moreover, must involve only the operations of addition, subtraction, multiplication, and division, together with the extraction of roots, since that was all that had been required for the solution of quadratic, cubic, and quartic equations. If such a formula were to exist, the quintic would accordingly be said to be solvable by radicals.

In 1770 Lagrange had analyzed all the successful methods he knew for equations of degrees 2, 3, and 4, in an attempt to see why they worked and how they could be generalized. His analysis of the problem in terms of permutations of the roots was promising, but he became more and more doubtful as the years went by that his complicated line of attack could be carried through. The first valid proof that the general quintic is not solvable by radicals was offered only after his death, in a startlingly short paper by Abel, written in 1824.

Abel also showed by example that some quintic equations were solvable by radicals and that some equations could be solved unexpectedly easily. For example, the equation x^{5} - 1 = 0 has one root x = 1, but the remaining four roots can be found just by extracting square roots, not fourth roots as might be expected. He therefore raised the question, “What equations of degree higher than 4 are solvable by radicals?”

Abel died in 1829 at the age of 26 and did not resolve the problem he had posed. Almost at once, however, the astonishing prodigy Évariste Galois burst upon the Parisian mathematical scene. He submitted an account of his novel theory of equations to the Academy of Sciences in 1829, but the manuscript was lost. A second version was also lost and was not found among Fourier's papers when Fourier, the secretary of the academy, died in 1830. Galois was killed in a duel in 1832, at the age of 20, and it was not until his papers were published in Joseph Liouville's Journal de mathématiques in 1846 that his work began to receive the attention it deserved.

His theory eventually. Galois emphasized the group (as he called it) of permutations of the roots of an equation. This move took him away from the equations themselves, instead turning toward the markedly more tractable study of permutations. To any given equation there corresponds a definite group, with a definite collection of subgroups. To explain which equations were solvable by radicals and which were not, Galois analyzed the ways in which these subgroups were related to one another: solvable equations gave rise to what are now called a chain of normal subgroups with cyclic quotients. This technical condition makes it clear how far mathematicians had gone from the familiar questions of 18th-century mathematics, and it marks a transition characteristic of modern mathematics: the replacement of formal calculation by conceptual analysis. This is a luxury available to the pure mathematician that the applied mathematician faced with a concrete problem cannot always afford.

According to this theory, a group is a set of objects that one can combine in pairs in such a way that the resulting object is also in the set. Moreover, this way of combination has to obey the following rules (here objects in the group are denoted a, b, etc., and the combination of a and b is written a * b):

There is an element, e, such that a * e = a = e * a for every element a in the group. This element is called the identity element of the group.

For every element a there is an element, written a^{-1}, with the property that a * a^{-1} = e = a^{-1} * a. The element a^{-1} is called the inverse of a.

For every a, b, and c in the group the associative law holds: (a * b) * c = a * (b * c).

Examples of groups include the integers with * interpreted as addition and the positive rational numbers with * interpreted as multiplication. An important property shared by some groups but not all is commutativity: for every element a and b, a * b = b * a. The rotations of an object in the plane around a fixed point form a commutative group, but the rotations of a three-dimensional object around a fixed point form a noncommutative group.

II. Read the paragraph speaking about how Galois' theory made the theory of equations into a part of the theory of groups and speak about it.

III. Read the last paragraph and speak about a group.