You may have seen that I posted this proof with some questions earlier today. But I found the answer to most of them. Now I have just one question regarding this proof, so I thought it would be better to delete the earlier post and create a new one, since the earlier one didn't get any answers.
The proof looks big, and the picture is big, but the author says that it is written in very much detail. My question will revolve around the red line, and I will post it after the picture:
My question is just how does he knows that $\Omega$ has enough elements to do what he does the over the red line? I get why it has enough elements to contain every algebraic extension of F, since an element of an algebraic extension of F, has to be a zero of a polynomial in $F[x]$, and he has made it so that we have enough elements for all possible zeros of all possible polynomials in $F[x]$.
But in the last part of the proof. How does he know that $\Omega$ has enough elements to construct $\bar{F}(\omega)$? I mean, he said that $\Omega$ was strictly bigger than A, so we know that we have more elements than we need for all possible algebraic extensions of F. But "strictly more" and "many more" is not precise?
Basically: A was created so that it has enough elements for all algebraic extensions of F, but how is it certain that $\Omega$ has enough elements for even one algebraic extension of $\bar{F}$ (the extension $\bar{F}(\omega)$)?
I must admit that I don't even know why the statement "Since $\Omega$ has many more elements than $\bar{F}$" is true. I know that Zorn's lemma uses the word maximal element, but by definition all we require is that a maximal element is not the entire set, so what is from stopping it from just being the entire set -1 element, er the entire set-"a very small subset"?
UPDATE:
I thought of an argument, do you think this works? Wikipedia talks about cantors theorem and a strict subset by taking the set of all subsets, is what I write under correct, and can be put in the proof?
We assume that $\Omega$ is created by taking the union of "all subsets of A" and "one more element" . Then we have all the A elements embedded in $\Omega$ as the singletons. Since $\bar{F}$ is an algebraic extension it is represented by these singletons.
Now we do as in the proof assume that $f(x)$ is a nonconstant polynomial in $\bar{F}[x]$, by Kroneker's theorem, there exists an extension field K, with an element s. This element is algebraic over $\bar{F}$ since it is a zero of a polynomial in $\bar{F}[x]$. So the set $F(s)$ is created by $k_0+…+k_{n-1}s^{n-1}$ for $k_i \in \bar{F}$. We can associate s with the "one more element", and since $\bar{F}$ is an algebraic extension, every k is associated with the singletons, and every $k_0+…+k_{n-1}s^{n-1}$ is naturally associated with the subset that is the union of the singletons that each k is associated with.
Hence we have enough elements. Does this work?
Best Answer
If $f(x)$ is a nonconstant polynomial in $\bar{F}[x]$ having no roots, the field $K=\bar{F}[x]/(f(x))$ is a proper algebraic extension of $\bar{F}$ where $f(x)$ has a root.
Now $|K\setminus\bar{F}|\le|K|<|\Omega|$ and $|\Omega\setminus\bar{F}|=|\Omega|$ by cardinal arithmetic. So you can find an injective map $$ \varphi\colon K\to\Omega $$ such that $\varphi(a)=a$ for $a\in\bar{F}$. You can transport the structure of $K$ on $\varphi(K)$ which so becomes a proper field extension of $\bar{F}$ contained in $\Omega$, contradicting the maximality of $\bar{F}$.