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When proving that rational numbers can be written in their lowest terms, we used the well-ordering principle to choose a numerator that is the smallest and then it contradicts the fact that this numerator can be further reduced. However, if we were given a general unique factorization domain (UFD), how are we suppose to show that every element in the field of fractions can still be expressed in the lowest terms (i.e the gcd(a,b) is not a unit).

I was thinking of going along the same lines as with the rational numbers case but now that I have the set of numerators how do I choose the "smallest" element in this case? Do I still apply the well-ordering principle, but then how can I say that an element is smaller than another in a general ring?

AMX
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  • Simply cancel their gcd, yielding a fraction with coprime elements (by the linked gcd distributive law). That works in any domain where all gcds exist (not only UFDs). – Bill Dubuque Feb 09 '23 at 00:00
  • @BillDubuque ah thanks so instead of proving via contradiction I just immediately show that the element can be written in terms that are coprime – AMX Feb 09 '23 at 00:19
  • You could do it by continually cancelling factors and using that divisibility is well-founded in a UFD, but cancelling their gcd is much easier. – Bill Dubuque Feb 09 '23 at 00:23
  • Alright, Thank you – AMX Feb 09 '23 at 00:49
  • e.g. choose a denom $d$ with least #prime factors. If the fraction wasn't reduced we could cancel another proper factor, yielding a denom with fewer prime factors than $d$, contra minimality of $,d\ \ $ – Bill Dubuque Feb 09 '23 at 00:53

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