Field of Fractions:
The field of fractions is a concept in abstract algebra and ring theory. It is used to construct a field from an integral domain, allowing division of elements that may not be formally fractions.
Formally, given an integral domain, which is a commutative ring where every non-zero element has a multiplicative inverse, the field of fractions is constructed by considering all possible ratios, or "fractions," of elements in the integral domain.
In the field of fractions, each element is represented as a fraction in the form of a/b, where a and b are elements of the integral domain and b is not equal to zero. This representation allows for the division of elements, even when they are not naturally fractions.
The field of fractions inherits the addition and multiplication operations from the integral domain. Addition is performed by adding the numerators and denominators separately, while multiplication is done by multiplying the numerators and denominators. Division in the field of fractions is defined as the multiplication of the numerator by the multiplicative inverse of the denominator.
The field of fractions has several important properties. It is a field, meaning that it satisfies all the axioms of a field, such as closure under addition and multiplication, and the existence of additive and multiplicative inverses. It is also the smallest field containing the integral domain, meaning that any other field containing the integral domain must contain the field of fractions as a subfield.
