I am reading "Higher Algebra" by A. Kurosh.
Let there be given in the field $P$ a subfield $P^{'}$ and an element $c$ exterior to $P^{'}$ and suppose we have a minimum subfield $P^{''}$ of $P$ which contains both $P^{'}$ and $c$. There can only be one such minimum subfield, since if $P^{'''}$ were one more subfield with these properties, then the intersection of subfields $P^{''}$ and $P^{'''}$ (i.e., the collection of elements common to both fields) would contain $P^{'}$ and the element $c$ and, together with any two of its elements, it would contain their sum (this sum must lie both in $P^{''}$ and in $P^{'''}$, and so also in their intersection) and likewise their product, difference and quotient; in other words the intersection would itself be a subfield, but this contradicts the minimality of the subfield $P^{''}$. We will say that the field $P^{''}$ is obtained by adjoining an element $c$ to the field $P^{'}$; symbolically, we write $P^{''}=P^{'}(c)$.
The author didn't define what a minimum subfield of $P$ which contains both $P^{'}$ and $c$ is.
I guess a subfield $P^{''}$ of $P$ is a minimum subfield of $P$ which contains both $P^{'}$ and $c$ if and only if for any subfield $P^{'''}$ of $P$ which contains both $P^{'}$ and $c$, $P^{''}\subset P^{'''}$ holds.
If we adopt this definition, then I think the uniqueness of a minimum subfield of $P$ which contains both $P^{'}$ and $c$ is obvious.
Why did the author write such a long proof?
Any reasonable reason?
By the way, when we prove the existence of such a minimum subfield, I think we need the similar argument which the author wrote for the uniqueness of a minimum subfield.
I think the intended definition of "minimal" here is "doesn't contain any smaller one". Then it requires proof that there aren't two distinct minimal ones. By way of analogy, there are lots of minimal nonempty sets (the singleton sets).