Confused about the definition of a Field Extension

Question

A particular author defines a Field Extension as a monomorphism (to be more detailed, as an injective homomorphism) between two Fields.

However, my idea of a Field Extension is that of a pair of Fields in which the 'smaller' Field is contained in the 'larger' one; in the sense that each element of the 'smaller' field is in the 'larger' one.

For me, a monomorphism seems different that a Field Extension. Am I missing something?Is the author maybe defining something else? Is my idea of a monomorphism wrong?

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I would really appreciate any help/thoughts!

Answer 1

If $i: F \to F'$ is a monomorphism of fields, then $i[F]$ is a subfield of $F'$ and is isomorphic to $F$. So both views come down to the same thing, as long as we don't care about differences between isomorphic fields.

Answer 2

If $F$ is a field and $K$ is a subfield, then there is a natural monomorphism $\iota\colon K\longrightarrow F$; it is defined by $\iota(x)=x$.

And if $\iota$ is a monomorphism from a field $K$ into a field $F$, then $F$ has a subfield isomorphic to $K$, which is $\iota(K)$.