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Looking at prime field, I notice that $\mathbb{Q}$ is not listed as an example. Is that not usually considered a prime field?
If the prime fields are to be considered a weakly initial set for the category of fields (as mentioned at field), then it needs to be included. I think we can also describe a prime field as a field admitting an epi from the initial object $\mathbb{Z}$ in the category of commutative rings. Or, as a residue field of a localization of $\mathbb{Z}$ with respect to a prime ideal (the case of $\mathbb{Q}$ corresponding to the zero ideal being a prime ideal).
Lang [Algebra, 3rd ed.] writes:
If $K$ is a field, then $K$ has characteristic $0$ or $p \gt 0$. In the first case, $K$ contains as a subfield an isomorphic image of the rational numbers, and in the second case, it contains an isomorphic image of $\mathbb{F}_p$. In either case, this subfield will be called the prime field (contained in $K$).
So it appears to me that $\mathbb{Q}$ is supposed to be a prime field.
I thought it was too.
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