Does the degree of a field extension depend on the embedding of the base field?

Question

To formulate the question more precisely, let $f$ be a field monomorphism from $F$ to $E$. The extension field $E$ can be considered a vector space over $F$, if scalar multiplication is defined by the equality $c\alpha=f(c)\alpha$, where $c$ is a scalar, $\alpha$ a vector, the multiplication in the left-hand side is that being defined, and the multiplication in the right is that under which $E$ is a field; call this scalar multiplication induced by $f$. If $f_1$ and $f_2$ are field monomorphisms from $F$ to $E$, then is the dimension of $E$ as a vector space with scalar multiplication induced by $f_1$ the same as that of $E$ as a vector space with scalar multiplication induced by $f_2$?

Answer 1

Let $F = \mathbf{C}(t)$ be the field of complex rational functions.

Also, let $E = \mathbf{C}(x)$ be the field of complex rational functions.

The embedding $t \mapsto x$ gives a field extension of degree $1$. The embedding $t \to x^2$ gives a field extension of degree $2$.