Why aren't there any coproducts in the category of $\bf{Fields}$? Well the question is stated in the title. 
I dont know much about field theory and i was suprised when i read it on wikipedia
please provide some examples
thanks in advance
 A: 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 $\mathbb{Q}(\sqrt{2}) \otimes_{\mathbb{Q}} \mathbb{Q}(\sqrt{3}) = \mathbb{Q}(\sqrt{2}, \sqrt{3})$ 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.
A: 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.)
