# Is the category of fields small?

I know that the category of groups $$\textbf{(Group)}$$ and rings $$\textbf{(Ring)}$$ are both only locally small, since any non-empty set can be made into a group or a ring.

However, when this comes to fields, I'm not sure if there's also an explicit construction making an infinite set to a field (we know not all finite set can be a field). Therefore I cannot confirm if the category of fields is small.

Thanks for any suggestion in advance.

• Can't an arbitrary countable set be viewed as $\mathbb{Q}?$ Or an uncountable one as $\mathbb{R}?$ Not at all sure if this helps... – green frog Dec 6 '18 at 10:04
• Well, I doubt if what you state is correct. On the other hand, even if it is correct, can it confirm that $\textbf{(Field)}$ is not small? – User Dec 6 '18 at 10:07
• Doubt no more. @伽罗瓦 is correct about the countable one. With the uncountable one, of course, one needs to be more careful. – Asaf Karagila Dec 6 '18 at 10:07
• Of course the countable one is trivial, but the uncountable is not correct, and this still doesn't solve my problem... – User Dec 6 '18 at 10:09
• For $\kappa$ large enough, you can biject $2^\kappa$ with the transcendence degree $\kappa$ extension of $\mathbb{F}_2$. So the category (Fields) is not small. – user10354138 Dec 6 '18 at 10:10

Let's fix the rational numbers. Right? That's just one set now. Given any other set which is disjoint from $$\Bbb Q$$, consider the transcendental extension $$\Bbb Q(A)$$, as a field. This alone shows that every set can be embedded into a field. And easy cardinality check will show that $$|A|=|\Bbb Q(A)|$$ whenever $$A$$ is infinite.
In particular, via transport of structure any infinite set can be made into a field. And so far we're only talking about purely transcendental extensions of $$\Bbb Q$$. This can be repeated with any field instead of $$\Bbb Q$$ (e.g. $$\Bbb F_2$$) as well.
• @伽罗瓦 Let $S$ be the class of all singleton sets (sets with just one elements). $S$ is a proper class. Why? For any set $X$, $\{X\}\in S$, so there is an injective function from the class of all sets into $S$. Now the subcategory of $\textsf{Sets}$ with objects $S$ is not small. – Alex Kruckman Dec 7 '18 at 19:26
• @伽罗瓦 Maybe you're thinking of "essentially small". For many kinds of concrete categories $C$, if $C$ doesn't contain a set of every cardinality, then $C$ is essentially small. – Alex Kruckman Dec 7 '18 at 19:28