In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0. The field of fractions of R is sometimes denoted by Quot(R) or Frac(R).
Mathematicians refer to this construction as the quotient field, field of fractions, or fraction field. All three are in common usage, and which is used is a matter of personal taste. The expression "quotient field" may sometimes run the risk of confusion with the quotient of a ring by an ideal, which is a quite different concept.
A multiplicative identity is not required for the role of the integral domain; this construction can be applied to any non-trivial commutative pseudo-ring with no zero divisors.
Read more about Field Of Fractions: Examples, Construction
Famous quotes containing the words field of and/or field:
“Never in the field of human conflict was so much owed by so many to so few.”
—Winston Churchill (1874–1965)
“Hardly a book of human worth, be it heaven’s own secret, is honestly placed before the reader; it is either shunned, given a Periclean funeral oration in a hundred and fifty words, or interred in the potter’s field of the newspapers’ back pages.”
—Edward Dahlberg (1900–1977)