What is the smallest infinite field?

Question

The real numbers and the rational numbers are both fields, but what is the smallest field. Is the set of rational numbers smaller than the set of reals, and if so is there a 'smaller' infinite set which is a field?

Answer 1

A plausible definition of a smallest infinite field would be a field that is isomorphic to a subfield of every other infinite field. With that definition, there is no smallest infinite field, because if two fields have different characteristics, one can't be a subfield of the other.

So might perhaps hope for a smallest subfield if one fixes the characteristic. If $\mathrm{char} = 0$, then this works, because any characeteristic zero field contains $\Bbb Q$. But if the characteristic is $p>0$, then this doesn't work. There's a smallest field of characteristic $p$, namely $\Bbb F_p$, but no smallest infinite field. One way to see this is to note that the algebraic closure $\overline{\Bbb F_p}$ of a finite field and the field of rational functions $\Bbb F_p(t)$ have no infinite subfield in common.

Answer 2

Any field for which $1, 2, 3, \dots$ are all distinct automatically has a subfield isomorphic to $\mathbb{Q}$. However, consider the field $F$ of quotients of polynomials with coefficients in $\mathbb{Z}/2\mathbb{Z}$. $F$ does not have a subfield isomorphic to $\mathbb{Q}$, because if it did, then that would imply that $1, 2, 3, \dots$ are all distinct in $F$.

Answer 3

As far as I know there's no total order on the category of fields. Normally some fields are not comparable as sets. Thus the notion of a smallest field does not quite make sense.

If you're looking for the infinite field of smallest order, there are quite a few, starting with the rationals, and other countable fields, like number fields, which are finite extensions of $\Bbb Q$. Of course there are others, like the fields of fractions of countable integral domains.