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Since residue classes modulo $p$ are not numbers but sets of numbers with several operations defined on them, i don't think we can compare them at all. Yet i'm not sure.

For sake of an example lets take $p = 2$. We can claim that,

$$[0]_2 < [1]_2.$$

But wouldn't that mean any odd number is greater than any even number? Which is obviously false...

After all, if we generalize this question, does an ordered finite field exist?

İbrahim İpek
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    If we want $a,b\ge 0\implies a+b\ge 0$, then $0$ can't be the sum of positive elements, therefore a field (or ring) of finite characteristic can't have a compatible order defined. – Berci Jul 25 '19 at 23:20
  • @Berci Why are you answering in a comment? – Arthur Jul 25 '19 at 23:54
  • I was lazy to look for a link of duplicate.. – Berci Jul 25 '19 at 23:55
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    Although finite fields are never ordered fields, your reasoning in the case of $p=2$ was incorrect. You wrote: "But wouldn't that mean that any odd number is greater than any even number?" No, it wouldn't mean that at all. The order relation in a field need not have anything to do with the usual ordering relation on the numbers that may be elements of elements of the field. – Andreas Blass Jul 26 '19 at 00:14
  • @Arthur It is obvious that the question is a duplicate, and repeating the same banalities as answers does not really improve the site. A comment is fine. – Jyrki Lahtonen Jul 27 '19 at 05:37

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