Definition: rational prime, rational integer
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.
$\endgroup$ 51 Answer
$\begingroup$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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