Specified extension field K/F

A specified extension field $K/F$ is a field extension with a fixed injective unital homomorphism $\iota:F\hookrightarrow K$ . The catalogue object is the field $K$ , together with this scalar inclusion; its $F$ -action is $a\cdot x=\iota(a)x$ .

Two vector-space views

The field is one-dimensional over itself. Its dimension over $F$ is $[K:F]$ , possibly infinite. The $K$ -linear endomorphisms are multiplications by elements of $K$ ; $F$ -linear maps have fewer scalar constraints and may form a much larger ring.

Two algebra views

The only unital $K$ -algebra endomorphism of $K$ is the identity, since every element is a scalar. Its unital $F$ -algebra automorphisms are precisely field automorphisms fixing $\iota(F)$ pointwise. This distinction depends on the specified inclusion, not on the carrier set alone.