Abstract Algebra

What is a field?

A field is an algebraic structure that shares the common rules for operations (addition, subtraction, multiplication, and division, except division by zero) of the rational, real, and complex numbers (but not integers, see below under “ring”). A field must have two operations, must have at least two elements, and must be commutative, distributive, and associative (see above for definitions). Formerly called “rational domain,” a field in both French (corps) and German (Körper) appropriately means “body.” A field with a finite number of members is called a Galois or finite field. Fields are useful to define such concepts as vectors and matrices.


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