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## Abstract Algebra - Free Harvard Courses

Toggle navigation Additional Book Information. Summary The new edition of Abstract Algebra: An Interactive Approach presents a hands-on and traditional approach to learning groups, rings, and fields.

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Reviews Praise for previous editions: "The textbook gives an introduction to algebra. Request an e-inspection copy. Share this Title. Related Titles. Shopping Cart Summary. Items Subtotal. View Cart. Algebraic structures, with their associated homomorphisms , form mathematical categories.

## Abstract Algebra

Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called variety of groups.

As in other parts of mathematics, concrete problems and examples have played important roles in the development of abstract algebra. Major themes include:.

Numerous textbooks in abstract algebra start with axiomatic definitions of various algebraic structures and then proceed to establish their properties. This creates a false impression that in algebra axioms had come first and then served as a motivation and as a basis of further study.

The true order of historical development was almost exactly the opposite. For example, the hypercomplex numbers of the nineteenth century had kinematic and physical motivations but challenged comprehension. Most theories that are now recognized as parts of algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts.

An archetypical example of this progressive synthesis can be seen in the history of group theory. There were several threads in the early development of group theory, in modern language loosely corresponding to number theory , theory of equations , and geometry. Leonhard Euler considered algebraic operations on numbers modulo an integer, modular arithmetic , in his generalization of Fermat's little theorem.

These investigations were taken much further by Carl Friedrich Gauss , who considered the structure of multiplicative groups of residues mod n and established many properties of cyclic and more general abelian groups that arise in this way. In his investigations of composition of binary quadratic forms , Gauss explicitly stated the associative law for the composition of forms, but like Euler before him, he seems to have been more interested in concrete results than in general theory.

In , Leopold Kronecker gave a definition of an abelian group in the context of ideal class groups of a number field, generalizing Gauss's work; but it appears he did not tie his definition with previous work on groups, particularly permutation groups. In , considering the same question, Heinrich M. Weber realized the connection and gave a similar definition that involved the cancellation property but omitted the existence of the inverse element , which was sufficient in his context finite groups.

Lagrange's goal was to understand why equations of third and fourth degree admit formulas for solutions, and he identified as key objects permutations of the roots. An important novel step taken by Lagrange in this paper was the abstract view of the roots, i. However, he did not consider composition of permutations.

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Serendipitously, the first edition of Edward Waring 's Meditationes Algebraicae Meditations on Algebra appeared in the same year, with an expanded version published in Waring proved the fundamental theorem of symmetric polynomials , and specially considered the relation between the roots of a quartic equation and its resolvent cubic. Kronecker claimed in that the study of modern algebra began with this first paper of Vandermonde.

Cauchy states quite clearly that Vandermonde had priority over Lagrange for this remarkable idea, which eventually led to the study of group theory. Paolo Ruffini was the first person to develop the theory of permutation groups , and like his predecessors, also in the context of solving algebraic equations. His goal was to establish the impossibility of an algebraic solution to a general algebraic equation of degree greater than four.

En route to this goal he introduced the notion of the order of an element of a group, conjugacy, the cycle decomposition of elements of permutation groups and the notions of primitive and imprimitive and proved some important theorems relating these concepts, such as.

Note, however, that he got by without formalizing the concept of a group, or even of a permutation group. The theory of permutation groups received further far-reaching development in the hands of Augustin Cauchy and Camille Jordan , both through introduction of new concepts and, primarily, a great wealth of results about special classes of permutation groups and even some general theorems.

Among other things, Jordan defined a notion of isomorphism , still in the context of permutation groups and, incidentally, it was he who put the term group in wide use.

The abstract notion of a group appeared for the first time in Arthur Cayley 's papers in Cayley realized that a group need not be a permutation group or even finite , and may instead consist of matrices , whose algebraic properties, such as multiplication and inverses, he systematically investigated in succeeding years. Much later Cayley would revisit the question whether abstract groups were more general than permutation groups, and establish that, in fact, any group is isomorphic to a group of permutations.

The end of the 19th and the beginning of the 20th century saw a tremendous shift in the methodology of mathematics. Abstract algebra emerged around the start of the 20th century, under the name modern algebra. Its study was part of the drive for more intellectual rigor in mathematics. Initially, the assumptions in classical algebra , on which the whole of mathematics and major parts of the natural sciences depend, took the form of axiomatic systems.

No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. Formal definitions of certain algebraic structures began to emerge in the 19th century. For example, results about various groups of permutations came to be seen as instances of general theorems that concern a general notion of an abstract group.

Questions of structure and classification of various mathematical objects came to forefront. These processes were occurring throughout all of mathematics, but became especially pronounced in algebra. Formal definition through primitive operations and axioms were proposed for many basic algebraic structures, such as groups , rings , and fields. Hence such things as group theory and ring theory took their places in pure mathematics. The algebraic investigations of general fields by Ernst Steinitz and of commutative and then general rings by David Hilbert , Emil Artin and Emmy Noether , building up on the work of Ernst Kummer , Leopold Kronecker and Richard Dedekind , who had considered ideals in commutative rings, and of Georg Frobenius and Issai Schur , concerning representation theory of groups, came to define abstract algebra.