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Abstract algebra is the field of mathematics concerned
with the study of algebraic structures such as groups, rings and fields. The term
"abstract algebra" is used to distinguish the field from "elementary algebra" or "high school algebra" which teaches the correct rules for manipulating formulas
and algebraic expressions involving real and complex numbers.
Historically, algebraic structures usually arose first in some other field of mathematics, were specified axiomatically, and
were then studied in their own right in abstract algebra. Because of this, abstract algebra has numerous fruitful connections to
all other branches of mathematics.
Examples of algebraic structures with a single binary operation
are:
More complicated examples include:
In universal algebra, all those definitions and facts are
collected that apply to all algebraic structures alike. All the above classes of objects, together with the proper notion of
homomorphism, form categories, and category theory frequently provides the formalism for translating between and comparing
different algebraic structures.
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