The term abstract algebra was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning.
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Abstract algebra came into existence during the nineteenth century as more complex problems and solution methods developed.
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Fully symbolic Abstract algebra did not appear until Francois Viete's 1591 New Algebra, and even this had some spelled out words that were given symbols in Descartes's 1637 La Geometrie.
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Abstract algebra distinguished a new symbolical algebra, distinct from the old arithmetical algebra.
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Whereas in arithmetical Abstract algebra is restricted to, in symbolical Abstract algebra all rules of operations hold with no restrictions.
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Abstract algebra defined nilpotent and idempotent elements and proved that any algebra contains one or the other.
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Abstract algebra further defined the discriminant of these forms, which is an invariant of a binary form.
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Abstract algebra extended this further in 1890 to Hilbert's basis theorem.
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Abstract algebra's definition was mainly the standard axioms: a set with two operations addition, which forms a group, and multiplication, which is associative, distributes over addition, and has an identity element.
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Abstract algebra axiomatically defined fields with the modern definition, classified them by their characteristic, and proved many theorems commonly seen today.
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Abstract algebra emerged around the start of the 20th century, under the name modern algebra.
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