Finite-field Hom-sets under field and ring conventions

In the category of fields, all maps preserve $1$ . The following are exact Hom-set calculations:

Source and target	Field homomorphisms	Arbitrary ring homomorphisms
$\mathbb F_2\to\mathbb F_4$	the unique prime-field inclusion	inclusion and zero
$\mathbb F_2\to\mathbb F_3$	none	zero only
$\mathbb F_4\to\mathbb F_8$	none	zero only
$\mathbb F_4\to\mathbb F_4$	identity and $x\mapsto x^2$	those two and zero

In the last row both field endomorphisms are automorphisms. “Zero only” is a singleton set, whereas “none” means the empty set.

Verification

A field homomorphism is injective and preserves characteristic, excluding $\mathbb F_2\to\mathbb F_3$ . A map $\mathbb F_4\to\mathbb F_8$ would inject a multiplicative group of order $3$ into one of order $7$ , contrary to Lagrange's theorem. Maps out of the prime field are determined by $1$ .

Write $\mathbb F_4=\mathbb F_2[u]/(u^2+u+1)$ . Its nontrivial element $u$ must map to one of the two roots $u,u^2$ , giving identity and Frobenius. Finally, every nonzero ring map between fields is unital: its value on $1$ is a nonzero idempotent. Thus allowing arbitrary ring maps adds precisely the zero map in each row.

Catalogue meaning

These are affirmative calculations of emptiness or nonemptiness. An unrecorded pair in the catalogue has status “not catalogued,” which is distinct from both outcomes above.