Context: In the first 4 pages of Neukirch's text algebraic number theory, there are references to 'rational primes' and 'rational integers'. These come up in the context of finding all primes and units in $\Bbb{Z}[i]$.

What does this refer to?

A guess: A rational prime in $\Bbb{Z}[i]$ is a prime element in $\Bbb{Z}[i]$, which is also an element of $\Bbb Q$, rather than being, say, $1+i$ (which is prime, but not a rational number). 'Rational integer' is less clear to me though, since all integers are rational.

Question: What do the terms 'rational integer' and 'rational prime' actually mean.

  • $\begingroup$ Algebraic integers (of which there are a surfeit in Neukirch's book) are not necessarily rational. $\endgroup$ – Angina Seng Feb 23 '18 at 5:48
  • $\begingroup$ @LordSharktheUnknown I haven't (explicitly(?)) mentioned algebraic integers? $\endgroup$ – user534352 Feb 23 '18 at 5:50
  • $\begingroup$ You wrote down $1+i$, which is an algebraic integer. $\endgroup$ – Angina Seng Feb 23 '18 at 5:51
  • $\begingroup$ @LordSharktheUnknown as an example of a prime element in $\Bbb{Z}[i]$ that isn't in $\Bbb Q$ (meaning what I assume is the sort of thing the term 'a rational prime' excludes intentionally). $\endgroup$ – user534352 Feb 23 '18 at 5:53
  • $\begingroup$ It is not true to say "all integers are rational" - it turns out to be useful to have a notion of "integer" in a wider context. Using the designation "rational integer" implies that you are in a context in which other integers may exist. $\endgroup$ – Mark Bennet Feb 23 '18 at 7:06

Usually "rational integer" and "rational prime" is terminology thrown around in algebraic number theory to mean integer of $\mathbb{Q}$ and prime of (the ring of integers of) $\mathbb{Q}$, i.e. elements of $\mathbb{Z}$ or primes in $\mathbb{Z}$. This terminology is used to differentiate between the term algebraic integer, which will often be said without the algebraic preceding it, and the term prime which might refer to a prime ideal in the ring of integers of a number field.


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