Abstract algebra was developed in the 19th century, deriving from the interest in solving equations, initially focusing on what is now called Galois theory, and on constructibility issues.
Josiah Willard Gibbs developed an algebra of vectors in three-dimensional space, and Arthur Cayley developed an algebra of matrices this is a noncommutative algebra. Some areas of mathematics that fall under the classification abstract algebra have the word algebra in their name; linear algebra is one example.
Others do not: group theory, ring theory, and field theory are examples. In this section, we list some areas of mathematics with the word 'algebra' in the name. Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic.
In algebra, numbers are often represented by symbols called variables such as a , n , x , y or z. This is useful because:. A polynomial is an expression that is the sum of a finite number of non-zero terms, each term consisting of the product of a constant and a finite number of variables raised to whole number powers. A polynomial expression is an expression that may be rewritten as a polynomial, by using commutativity, associativity and distributivity of addition and multiplication. A polynomial function is a function that is defined by a polynomial, or, equivalently, by a polynomial expression.
The two preceding examples define the same polynomial function. Two important and related problems in algebra are the factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials that can not be factored any further, and the computation of polynomial greatest common divisors. A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable. Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts.
Here are listed fundamental concepts in abstract algebra. Sets : Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets : a collection of all objects called elements selected by property specific for the set. All collections of the familiar types of numbers are sets. Set theory is a branch of logic and not technically a branch of algebra. The notion of binary operation is meaningless without the set on which the operation is defined.
Identity elements : The numbers zero and one are abstracted to give the notion of an identity element for an operation. Zero is the identity element for addition and one is the identity element for multiplication.
Not all sets and operator combinations have an identity element; for example, the set of positive natural numbers 1, 2, 3, Inverse elements : The negative numbers give rise to the concept of inverse elements. Associativity : Addition of integers has a property called associativity.
That is, the grouping of the numbers to be added does not affect the sum. This property is shared by most binary operations, but not subtraction or division or octonion multiplication. Commutativity : Addition and multiplication of real numbers are both commutative. That is, the order of the numbers does not affect the result. This property does not hold for all binary operations. For example, matrix multiplication and quaternion multiplication are both non-commutative.
Combining the above concepts gives one of the most important structures in mathematics: a group. For example, the set of integers under the operation of addition is a group. The non-zero rational numbers form a group under multiplication. This website uses cookies to improve your experience while you navigate through the website. Out of these cookies, the cookies that are categorized as necessary are stored on your browser as they are as essential for the working of basic functionalities of the website.
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Changes for the ninth edition include new exercises, new examples, new biographies, new quotes, new appliactions, and a freshening of the historical notes and biographies from the 8th edition. The website www. Additionally, Cengage offers a Student Solutions Manual, available for purchase separately, with detailed solutions to the odd-numbered exercises in the book ISBN ; ISBN I wish to thank Roger Lipsett for serving as accuracy checker and my UMD colleague Robert McFarland for giving me a number of exercises that are in this edition.
Finally, I am grateful for the help of Manoj Chander and the composition work of the whole team at Lumina Datamatics. Do you like this book? Please share with your friends, let's read it!! Search Ebook here:. Book Preface Although I wrote the first edition of this book more than thirty years ago, my goals for it remain the same.
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