### 数学代写|抽象代数作业代写abstract algebra代考|Sources of group theory

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## 数学代写|抽象代数作业代写abstract algebra代考|Classical Algebra

The major problems in algebra at the time $(1770)$ that Lagrange wrote his fundamental memoir “Reflections on the solution of algebraic equations” concerned polynomial equations. There were “theoretical” questions dealing with the existence and nature of the roots-for example, does every equation have a root? how many roots are there? are they real, complex, positive, negative?-and “practical” questions dealing with methods for finding the roots. In the latter instance there were exact methods and approximate methods. In what follows we mention exact methods.

The Babylonians knew how to solve quadratic equations, essentially by the method of completing the square, around $1600 \mathrm{BC}$ (see Chapter 1). Algebraic methods for solving the cubic and the quartic were given around 1540 (Chapter 1). One of the major problems for the next two centuries was the algebraic solution of the quintic. This is the task Lagrange set for himself in his paper of 1770 .In this paper Lagrange first analyzed the various known methods, devised by Viète, Descartes, Euler, and Bezout, for solving cubic and quartic equations. He showed that the common feature of these methods is the reduction of such equations to auxiliary equations-the so-called resolvent equations. The latter are one degree lower than the original equations.

Lagrange next attempted a similar analysis of polynomial equations of arbitrary degree $n$. With each such equation he associated a resolvent equation, as follows: let $f(x)$ be the original equation, with roots $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$. Pick a rational function $\mathbf{R}\left(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\right)$ of the roots and coefficients of $f(x)$. (Lagrange described methods for doing this.) Consider the different values which $\mathbf{R}\left(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\right)$ assumes under all the $n$ ! permutations of the roots $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$ of $f(x)$. If these are denoted by $y_{1}, y_{2}, y_{3}, \ldots, y_{k}$, then the resolvent equation is given by $g(x)=\left(x-y_{1}\right)\left(x-y_{2}\right) \cdots\left(x-y_{k}\right) .$

It is imporotant to note that the coefficients of $g(x)$ aree symmetric functions in $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$, hence they are polynomials in the elementary symmetric functions of $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$; that is, they are polynomials in the coefficients of the original equation $f(x)$. Lagrange showed that $k$ divides $n$ !-the source of what we call Lagrange’s theorem in group theory.

For example, if $f(x)$ is a quartic with roots $x_{1}, x_{2}, x_{3}, x_{4}$, then $\mathbf{R}\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ may be taken to be $x_{1} x_{2}+x_{3} x_{4}$, and this function assumes three distinct values under the twenty-four permutations of $x_{1}, x_{2}, x_{3}, x_{4}$. Thus the resolvent equation of a quartic is a cubic. However, in carrying over this analysis to the quintic Lagrange found that the resolvent equation is of degree six.

Although Lagrange did not succeed in resolving the problem of the algebraic solvability of the quintic, his work was a milestone. It was the first time that an association was made between the solutions of a polynomial equation and the permutations of its roots. In fact, the study of the permutations of the roots of an equation was a cornerstone of Lagrange’s general theory of algebraic equations. This, he speculated, formed “the true principles of the solution of equations.” He was, of course, vindicated in this by Galois. Although Lagrange spoke of permutations without considering a “calculus” of permutations (e.g., there is no consideration of their composition or closure), it can be said that the germ of the group concept-as a group of permutations-is present in his work. For details see [12], [16], [19], [25], [33].

## 数学代写|抽象代数作业代写abstract algebra代考|Number Theory

In the Disquisitiones Arithmeticae (Arithmetical Investigations) of 1801 Gauss summarized and unified much of the number theory that preceded him. The work also suggested new directions which kept mathematicians occupied for the entire century. As for its impact on group theory, the Disquisitiones may be said to have initiated the theory of finite abelian groups. In fact, Gauss established many of the significant properties of these groups without using any of the terminology of group theory.
The groups appeared in four different guises: the additive group of integers modulo $m$, the multiplicative group of integers relatively prime to $m$, modulo $m$, the group of equivalence classes of binary quadratic forms, and the group of $n$-th roots of unity. And although these examples turned up in number-theoretic contexts, it is as abelian groups that Gauss treated them, using what are clear prototypes of modern algebraic proofs.

For example, considering the nonzero integers modulo $p$ ( $p$ a prime), he showed that they are all powers of a single element; that is, that the group $Z_{p}^{*}$ of such integers

is cyclic. Moreover, he determined the number of generators of this group, showing that it is equal to $\varphi(p-1)$, where $\varphi$ is Euler’s $\varphi$-function.

Given any element of $Z_{p}^{}$, he defined the order of the element (without using the terminology) and showed that the order of an element is a divisor of $p-1$. He then used this result to prove Fermat’s “little theorem,” namely that $a^{p-1} \equiv 1(\bmod p)$ if $p$ does not divide $a$, thus employing group-theoretic ideas to prove number-theoretic results. Next he showed that if $t$ is a positive integer which divides $p-1$, then there exists an element in $Z_{p}^{}$ whose order is $t$-essentially the converse of Lagrange’s theorem for cyclic groups.

Concerning the $n$-th roots of 1 , which he considered in connection with the cyclotomic equation, he showed that they too form a cyclic group. In relation to this group he raised and answered many of the same questions he raised and answered in the case of $Z_{p}^{*}$.

The problem of representing integers by binary quadratic forms goes back to Fermat in the early seventeenth century. (Recall his theorem that every prime of the form $4 n+1$ can be represented as a sum of two squares $x^{2}+y^{2}$.) Gauss devoted a large part of the Disquisitiones to an exhaustive study of binary quadratic forms and the representation of integers by such forms.

A binary quadratic form is an expression of the form $a x^{2}+b x y+c y^{2}$, with $a, b, c$ integers. Gauss defined a composition on such forms, and remarked that if $K_{1}$ and $K_{2}$ are two such forms, one may denote their composition by $K_{1}+K_{2}$. He then showed that this composition is associative and commutative, that there exists an identity, and that each form has an inverse, thus verifying all the properties of an abelian group.

Despite these remarkable insights, one should not infer that Gauss had the concept of an abstract group, or even of a finite abelian group. Although the arguments in the Disquisitiones are quite general, each of the various types of “groups” he considered was dealt with separately-there was no unifying group-theoretic method which he applied to all cases.
For further details see [5], [9], [25], [30], [33].

## 数学代写|抽象代数作业代写abstract algebra代考|Geometry

We are referring here to Klein’s famous and influential (but see [18]) lecture entitled “A Comparative Review of Recent Researches in Geometry,” which he delivered in 1872 on the occasion of his admission to the faculty of the University of Erlangen. The aim of this so-called Erlangen Program was the classification of geometry as the study of invariants under various groups of transformations. Here there appear groups such as the projective group, the group of rigid motions, the group of similarities, the hyperbolic group, the elliptic groups, as well as the geometries associated with them. (The affine group was not mentioned by Klein.) Now for some background leading to Klein’s Erlangen Program.

The nineteenth century witnessed an explosive growth in geometry, both in scope and in depth. New geometries emerged: projective geometry, noneuclidean geometries, differential geometry, algebraic geometry, $n$-dimensional geometry, and

Grassmann’s geometry of extension. Various geometric methods competed for supremacy: the synthetic versus the analytic, the metric versus the projective.

At mid-century a major problem had arisen, namely the classification of the relations and inner connections among the different geometries and geometric methods. This gave rise to the study of “geometric relations,” focusing on the study of properties of figures invariant under transformations. Soon the focus shifted to a study of the transformations themselves. Thus the study of the geometric relations of figures became the study of the associated transformations.

Various types of transformations (e.g., collineations, circular transformations, inversive transformations, affinities) became the objects of specialized studies. Subsequently, the logical connections among transformations were investigated, and this led to the problem of classifying transformations, and eventually to Klein’s group-theoretic synthesis of geometry.

Klein’s use of groups in geometry was the final stage in bringing order to geometry. An intermediate stage was the founding of the first major theory of classification in geometry, beginning in the $1850 \mathrm{~s}$, the Cayley-Sylvester Invariant Theory. Here the objective was to study invariants of “forms” under transformations of their variables (see Chapter 8.1). This theory of classification, the precursor of Klein’s Erlangen Program, can be said to be implicitly group-theoretic. Klein’s use of groups in geometry was, of course, explicit. (For a thorough analysis of implicit group-theoretic thinking in geometry leading to Klein’s Erlangen Program see [33].)

In the next section we will note the significance of Klein’s Erlangen Program (and his other works) for the evolution of group theory. Since the Program originated a hundred years after Lagrange’s work and eighty years after Gauss’ work, its importance for group theory can best be appreciated after a discussion of the evolution of group theory beginning with the works of Lagrange and Gauss and ending with the period around 1870 .

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