Editing Field of fractions (section)

== Examples ==

* The field of fractions of the ring of [[Integer#Algebraic_properties|integers]] is the field of [[rational number|rationals]]: <math>\Q = \operatorname{Frac}(\Z)</math>.
* Let <math>R:=\{a+b\mathrm{i} \mid a,b \in \Z\}</math> be the ring of [[Gaussian integer]]s. Then <math>\operatorname{Frac}(R)=\{c+d\mathrm{i}\mid c,d\in\Q\}</math>, the field of [[Gaussian rational]]s.
* The field of fractions of a field is canonically [[isomorphism|isomorphic]] to the field itself.
* Given a field <math>K</math>, the field of fractions of the [[polynomial ring]] in one indeterminate <math>K[X]</math> (which is an integral domain), is called the ''{{visible anchor|field of rational functions}}'', ''field of rational fractions'', or ''field of rational expressions''<ref>{{cite book |first=Ėrnest Borisovich  |last=Vinberg |url=https://books.google.com/books?id=rzNq39lvNt0C&pg=PA132 |title=A course in algebra |year=2003 |page=131 |isbn=978-0-8218-8394-5 |publisher=American Mathematical Society}}</ref><ref>{{cite book|first=Stephan |last=Foldes|url=https://archive.org/details/fundamentalstruc0000fold|title=Fundamental structures of algebra and discrete mathematics |publisher=Wiley |year=1994|page=[https://archive.org/details/fundamentalstruc0000fold/page/128 128]|url-access=registration |isbn=0-471-57180-6}}</ref><ref>{{cite book |first=Pierre Antoine |last=Grillet |chapter=3.5 Rings: Polynomials in One Variable |chapter-url=https://books.google.com/books?id=LJtyhu8-xYwC&pg=PA124|title=Abstract algebra|year=2007|page=124 |isbn=978-0-387-71568-1 |publisher=Springer}}</ref><ref>{{cite book|last1 = Marecek | first1 = Lynn | last2 = Mathis | first2 = Andrea Honeycutt | title = Intermediate Algebra 2e | date = 6 May 2020 | publisher = [[OpenStax]] <!-- | location = Houston, Texas -->|  url = https://openstax.org/details/books/intermediate-algebra-2e | at = §7.1}}</ref> and is denoted <math>K(X)</math>.
* The field of fractions of the [[convolution]] ring of half-line functions yields a [[convolution quotient | space of operators]], including the [[Dirac delta function]], [[differential operator]], and [[integral operator]].  This construction gives an alternate representation of the [[Laplace transform]] that does not depend explicitly on an integral transform.<ref>{{cite book | first=Jan | last=Mikusiński | url=https://books.google.com/books?id=e8LSBQAAQBAJ | title=Operational Calculus| date=14 July 2014 | publisher=Elsevier | isbn=9781483278933 }}</ref>