Why does Fld not have an initial object?

Question

My Algebra book says that the category Fld of fields has no initial object.

Why would $\{0,1\}$ not be an initial object? Does it not have a unique homomorphism to every other field?

Answer 1

Any field fails to embed in fields of different characteristic. (Consider this a basic exercise.)

Answer 2

The obvious mapping would seem to be $0\mapsto0$ and $1\mapsto1$, but that implies $1+1\mapsto0$, and so $1+1$ would have no multiplicative inverse. A homomorphism would map inverses to inverses.

Answer 3

It is perhaps worth noting that, in some texts, the definition of a field does not require the additive and multiplicative identities to be distinct. It isn't difficult to verify that $\{0\}$ satisfies the field axioms if we remove this restriction, and in particular, such texts would consider $\{0\}$ to be precisely the initial object of $\mathbf{Fld}.$