Talk:Number Field has Rational Numbers as Subfield
In response to the comment which was included in the "summary" field of the last edit (which is the least useful place to open a discussion, BTW):
- " Couldn't we also have $\Z \subseteq F$ by repeatedly adding $1$ to itself then getting $\Q \subseteq F$ by multiplying through inverses?"
That is the approach taken by Rational Numbers form Prime Field, is it not? So yes, that is what this proof effectively does. What we have here is arguably more useful, because it provides a direct consequence of the primeness of $\Q$ as a field, rather than fail to make that connection, leaving the reader possibly unaware not only of the significance of $\Q$ being a prime field, but even unaware of the fact that $\Q$ is a prime field in the first place, or even the existence of the concept of "prime field" in the first place.
By all means enter a second proof which explicitly does that fiddly thing, which to be rigorous needs to be done formally by induction, but I would not replace the existing proof with it. --prime mover (talk) 09:32, 27 May 2021 (UTC)