My conjecture is motivated by the desire, given a field $F$, to find an extension field by "adjoining" elements. Let's say $\forall x\in F[x*x+1\neq0]$ and we want to "adjoin" an element $α$ satisfying $α*α+1=0$.
A typical abstract algebra book (like Artin, see p. 444) would probably accomplish this by defining the "extension field" $F(α):=F[x]/(x^2+1)$. The problem with this approach, however, is that we will not have $F(α)\supseteq F$.
Therefore, I propose the following conjecture and would like to know whether it is true or false.
Let $(F,+_F,*_F)$ be a field.
Then there exists a set $K\supseteq F$ and functions $+,*:K^2\to K$ satisfying the following properties:
(i) $+|_{F^2}=+_F$
(ii) $*|_{F^2}=*_F$
(iii) $(K,+,*)$ is a field
(iv) $\exists α\in K[α*α+1=0]$ (where 1 and 0 denote the multiplicative/additive identities of $F$)
I have included the set-theory tag because my question regards the existence of sets as described by the ZFC axioms.
You don't need to insist that a field extension $K/F$ actually satisfies $F \subseteq K$. It is enough that $F$ is embedded in $K$, via a ring homomorphism $F \to K$. This homomorphism must be injective because $F$ is a field. The theory of field extensions can be done beautifully and instructively using embeddings.
If you insist on strict inclusion, consider $K= (F \times F^*)\cup F$. Then $F \subseteq K$ and you can set $\alpha = (0,1)$. This construction is artificial, because it effectively identifies $F$ with the removed $F \times 0$.