Understanding Fields in Mathematics: Definition and Properties

A field is a set in which we can do addition, subtraction, multiplication and division without leaving the set.

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True. A field is a set of elements in which two operations, typically called addition and multiplication, are defined and satisfy certain axioms. These operations have certain properties that allow us to perform arithmetic operations within the set. Specifically, for a set to be considered a field, it must satisfy the following conditions:

1. Closure under addition and multiplication: If a and b are in the field, then a + b and a x b are also in the field.
2. Associativity of addition and multiplication: For any a, b, and c in the field, (a + b) + c = a + (b + c) and (a x b) x c = a x (b x c).
3. Commutativity of addition and multiplication: For any a and b in the field, a + b = b + a and a x b = b x a.
4. Existence of additive and multiplicative identities: There exist elements 0 and 1 in the field such that for any a in the field, a + 0 = a and a x 1 = a.
5. Existence of additive and multiplicative inverses: For any a in the field, there exists an element -a in the field such that a + (-a) = 0, and for any non-zero a in the field, there exists an element a^-1 in the field such that a x a^-1 = 1.
6. Distributivity of multiplication over addition: For any a, b, and c in the field, a x (b + c) = (a x b) + (a x c).

In summary, a field is a set that satisfies these conditions, which allows us to perform arithmetic operations within the set. Examples of fields include the set of rational numbers, the set of real numbers, and the set of complex numbers.

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