Why aren't there any coproducts in the category of $\bf{Fields}$?

Solution 1:

Among other things, a coproduct of objects $F_1$ and $F_2$ in a category is an object $F$ together with morphisms $\iota_1: F_1 \rightarrow F$, $\iota_2: F_2 \rightarrow F$.

In order to have a homomorphism between two fields $K$ and $L$, $K$ and $L$ must have the same characteristic. Thus for instance $\mathbb{Q}$ and $\mathbb{Z}/p\mathbb{Z}$ (for any prime $p$) cannot have a coproduct in the category of fields. (Added: they can't have a product either, for almost exactly the same reasons.)

Solution 2:

As Pete said, two fields $F_1$ and $F_2$ can only have a coproduct if they have the same prime field $F$ ($F=\mathbb Q$ or $F=\mathbb F_p$).

a) If the $F$-algebra $F_1\otimes_F F_2$ is a field, then it is a coproduct of $F_1$ and $F_2$ in the category of fields.
The simplest examples are the fields $\mathbb Q(\sqrt 2) \otimes_\mathbb Q (\sqrt 3)=\mathbb Q(\sqrt 2,\sqrt 3)$ and $\mathbb F_{p^2} \otimes_{\mathbb F_p} \mathbb F_{p^3}=\mathbb F_{p^6}$
It is however a delicate question to decide whether $F_1\otimes_F F_2$ is a field: see here for many examples and non-examples.

b) If $F_1\otimes_F F_2$ is not a field and if a coproduct $F_1\sqcup F_2$ of $F_1$ and $F_2$ exists, we have a ring morphism $F_1\otimes_F F_2\to F_1\sqcup F_2$. But I don't know if it really possible that $F_1\sqcup F_2$ exists if $F_1\otimes_F F_2$ is not a field .