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Let $K$ be an arbitrary number field and $\mathcal{O}_K$ its ring of integers.

I have seen many concrete examples about finding prime elements. For example I calculated the prime elements of $\mathbb{Z}[i]$, or $\mathbb{Z}[\sqrt{-2}]$.

But how does this works in general? Exist an algorithm, which finds (some of) the prime elements of $\mathcal{O}_K$? Moreover does every ring of integers of a number field contains at least one prime element?

Additionally what happens if I consider function fields instead of number fields?

Sqyuli
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  • You should clarify if you really want prime elements, prime ideals or irreducible elements. This three sets are the same for the examples you wrote, since they are PID. – xarles Nov 10 '18 at 18:09
  • I really want prime elements. – Sqyuli Nov 10 '18 at 18:20
  • With a properly defined norm function in a UFD, if $N(n)$ is prime in $\mathbb{Z}$, then $n$ is prime. For example, in $\mathbb{Z}[i]$ you see that $N(2 + i) = 5$, which is prime in $\mathbb{Z}$, while $N(5) = 25$. – Mr. Brooks Nov 10 '18 at 22:23
  • Yes, but this only works for a UFD, because in this case prime elements and irreducible elements are the same. In the general case I only get that if $N(n)$ is prime in $\mathbb{Z}$, then $n$ is irreducible. [And of course only if the field of integers looks like $\mathbb{Z}[\sqrt{d}]$] – Sqyuli Nov 10 '18 at 23:12

1 Answers1

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Hint concerning the "Moreover"-question: Try to use Euclid's argument to show for (some) $K$, that $\mathcal{O}_K$ has infinitely many prime elements. For quadratic number fields see

Infinitely many primes in the ring of integers for any quadratic field

Dietrich Burde
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