Abstract
The word algebra is part of the title of an Arabic manuscript from about 800 A.D. giving rules for solving equations, and until about 100 years ago algebra was just the theory of equations. Nowadays it is best described as dealing with more or less formal mathematical operations and relations. Modern algebra is really a collection of second generation abstract models drawn from many parts of mathematics. Economy of notation prescribes that new symbols should be avoided unless absolutely necessary. For this reason familiar operational signs, e.g., those for addition and multiplication, are used again and again but acquire new meanings depending on the model. The objects of algebra are classified by the kind of operations that can be performed in them. They carry names like “ring” and “ideal,” which may sound funny at first; but this feeling wears off with a closer acquaintance.
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Literature
Commutative algebra, touched upon here, is an important part of mathematics closely connected with algebraic geometry. The book Commutative Algebra, by Atiyah and MacDonald (Addison-Wesley, 1969) provides a close-up view of the subject. It is a bit tough but goes along at a brisk pace, contains plenty of exercises, and is also rather comprehensive.
A Survey of Modern Algebra, by Birkhoff and MacLane (Macmillan, 1963), and The Theory of Groups, by M. Hall (Macmillan, 1959) are both easy to read and contain a lot of material. Algebra, by S. Lang (Addison-Wesley, 1965) is a rather tough, encyclopedic book. Van der Waerden’s classic, Moderne Algebra (from 1932) is now available in paperback under the title of Algebra (Springer-Verlag, 1968).
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© 1977 Springer-Verlag New York Inc.
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Gårding, L. (1977). Algebra. In: Encounter with Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4615-9641-7_3
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DOI: https://doi.org/10.1007/978-1-4615-9641-7_3
Publisher Name: Springer, New York, NY
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