Editing Irreducible fraction (section)

==Generalization==

The notion of irreducible fraction generalizes to the [[field of fractions]] of any [[unique factorization domain]]: any element of such a field can be written as a fraction in which denominator and numerator are coprime, by dividing both by their greatest common divisor.<ref>{{citation|title=Abstract Algebra|first=Paul B.|last=Garrett|publisher=CRC Press|year=2007|isbn=9781584886907|page=183|url=https://books.google.com/books?id=CZzSBQAAQBAJ&pg=PA183}}.</ref> This applies notably to [[rational fraction|rational expressions]] over a field. The irreducible fraction for a given element is unique up to multiplication of denominator and numerator by the same invertible element. In the case of the rational numbers this means that any number has two irreducible fractions, related by a change of sign of both numerator and denominator; this ambiguity can be removed by requiring the denominator to be positive. In the case of rational functions the denominator could similarly be required to be a [[monic polynomial]].<ref>{{citation|title=Abstract Algebra|volume=242|series=Graduate Texts in Mathematics|first=Pierre Antoine|last=Grillet|publisher=Springer|year=2007|isbn=9780387715681|at=Lemma 9.2, p.&nbsp;183|url=https://books.google.com/books?id=CZzSBQAAQBAJ&pg=PA183}}.</ref>