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Abstract algebra

History[ edit ] As in other parts of mathematics, concrete problems and examples have played important roles in the development of abstract algebra. Major themes include: Solving of systems of linear equations, which led to linear algebra Attempts to find formulas for solutions of general polynomial equations of higher degree that resulted in discovery of groups as abstract manifestations of symmetry Arithmetical investigations of quadratic and higher degree forms and diophantine equations , that directly produced the notions of a ring and ideal.

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.

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An archetypical example of this progressive synthesis can be seen in the history of group theory. Early group theory[ edit ] 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.

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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.

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.

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 if G is a subgroup of S5 whose order is divisible by 5 then G contains an element of order 5.

Note, however, that he got by without formalizing the concept of a group, or even of a permutation group. Meet Standards Choose your state to see how Discovering Advanced Algebra correlates to your state standards. Choose Your State West Virginia Wisconsin Wyoming. Differentiating Instruction. What's NEW! Beyond the Printed Text!

The Discovering Advanced Algebra approach improves learning in many ways: Real data, real-life situations, and real-world applications will help you motivate students and show them the importance of what they are learning. The sequence of mathematical experiences aids your students in making conceptual connections between the many topics covered in Algebra 2.

Your students will develop and practice skills, such as multiplying and factoring polynomials and solving many kinds of equations in a wide variety of mathematical settings.

The text also provides regular opportunities for students to practice and review skills learned in previous algebra courses. Investigations, visual representations, and opportunities for discussion will enable you to integrate multiple teaching modes into your classroom so that visual, auditory, and kinesthetic learners all benefit. To help you reach all your students, every lesson has a Differentiating Instruction box with suggestions for modifying the lessons for ELL students, students who need extra support, and advanced students.An important novel step taken by Lagrange in this paper was the abstract view of the roots, i.