The fraction (rational number) a/b is said to be a reduced fraction (or in reduced form) if it cannot be further reduced, i.e. if there is no integer d>1 which divides both a and b (equivalently, GCD(a,b) = 1).

We think of reduced fractions as the "right way™" to write down a fraction: "1/3" is much nicer than "7/21", even though the two are equal. But note that "-1/-3" is also a reduced fraction! For the rational numbers we further reduce the number of ways to write down a fraction by demanding that the denominator be positive.

Whenever we are in the field of fractions of some Gaussian domain R, we can talk of reduced fractions. However, in general the field of fractions isn't an ordered field, so we will have multiple equal reduced fractions for every element: as many as there units, in fact.

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