Let $F$ be a field and let $f(x)$ be a non constant element of $F[x]$. Then, there exists a splitting field $E$ for $f(x)$ over $F$.
I have some queries regarding this theorem of existence of splitting fields.
Firstly, let $E$ be an extension field of $F$ and let $f(x) \in F[x]$. We say that $f(x)$ splits in $E$ if $f(x)$ can be factored as a product of linear factors in $E[x]$.
We call $E$ a splitting field for $f(x)$ over $F$ if $f(x)$ splits in $E$ but in no proper sub field of $E$
$(i)$ The above theorem must be true $\forall~~f(x)\in F[x]$ , but, there can exist $f(x)\in F[x]$ such that they split completely in $F$ ( and $F$ is clearly a sub field of $E$ ). Isn't this contradictory to the definition of a splitting field?
$(ii)$ The proof in my book is through induction. Is there a proof other than induction?
Thank you for your help.
(i) If $f$ splits in $F$ then its splitting field over $F$ is just $E=F$, so no issue.
(ii) Quotient $F[x]/(f(x))$ by a maximal ideal, which necessarily exists, to get a field with an element, the image of $x$ in the quotient, satisfying the polynomial relation $f(x)=0$. (One can justifiably argue this only pushes induction back to Zorn's lemma, but that's just where I like it to be, not with degrees.)