Category of Field has no initial object

Solution 1:

Note that in the category of all fields, given any field $\mathbf{F}$ there is always a different field $\mathbf{K}$ such that there are no field homomorphisms from $\mathbf{F}$ to $\mathbf{K}$ nor from $\mathbf{K}$ to $\mathbf{F}$. Therefore, there can be neither an initial nor a terminal object (because the former requires maps into every object, and the latter requires maps from every object).

And the reason this happens is that there can be no field homomorphisms between fields of different characteristics.

Now, if you change to the category of $\mathbf{Field}_{p}$, where $p$ is either a prime or $0$, and this is the category of all fields of characteristic $p$, then this category does have an initial object, namely the prime field of characteristic $p$: $\mathbf{F}_p$ for $p\gt 0$, and $\mathbb{Q}$ for $p=0$.