Let $\mathbb F$ be a field and let $a, b, c, d$ be fixed elements in the field $\mathbb F$.
Consider the formulas


1) $\exists\;x\;\;:\;\;x^2=-1.$ 


2) $\exists\;x\;\;:\;\;(xa=c\land xb=d).$


Formula $(1)$ can be false in $\mathbb F$ but true in a field extension of $\mathbb F$. For exemple, formula $(1)$ is false in the reals $\mathbb R$ but true in the complex numbers $\mathbb C$. 


The same does not happen with the formula $(2)$ since formula $(2)$ is true if and only if the matrix 
$$\left(\begin{array}{cc}a&b\\c&d\end{array}\right),$$
is singular, and it is well known that the singularity of a matrix is not changed by an extension of the field that contains the elements of the matrix.


My question is:


Is there any characterization of the formulas, in a field, whose validity does not change by an extension of the field?


Remark: I put this question yesterday in <https://math.stackexchange.com/questions/561178/formulas-in-a-field-and-in-a-field-extension>. But was not answered. 



Formulas in a Field and in a Field Extension
This is a major clean-up of my previous argument, thanks to the very helpful comments of Emil Jerabek. 


I interpret the question as follows: Characterize all sentences in the language of fields that are preserved under field extensions.


Claim: A formula $\phi$ is preserved under field extensions if and only if the field axioms prove that $\phi$ is equivalent to an existential formula.


For definiteness, we take the language of fields to have operations for addition and multiplication, and distinguished elements for 0 and 1.


The main tool to establish the claim is the following 


Lemma: Suppose $M$ is a structure in any old language, and $T$ is a theory in that language. Let $\forall(T)$ denote the set of universal consequences of $T$. If $M\models\forall(T)$ then $M$ extends to a model of $T$.


Proof-sketch: The diagram of $M$ cannot be inconsistent with $T$, because otherwise $T$ would imply the negation of some formula in the diagram of $M$, and therefore would imply the universal closure of the negation of that formula, contrary to the assumption $M\models\forall(T)$. But then any model of the diagram of $M$ plus $T$ will do the job we require. 


Now we prove the Claim. The non-trivial part is to show that if the theory of fields (which we will henceforth call $T$) cannot prove the equivalence between $\phi$ and any existential sentence, then $\phi$ is not preserved under extensions. 


But our premiss is equivalent to the statement that $T$ cannot prove the equivalence between $\neg\phi$ and any universal sentence. Let $S$ be the set of universal consequences of $T+\neg\phi$. Then $S+T\nvdash\neg\phi$. For otherwise, using the fact that $S$ is closed under conjunction (at least up to logical equivalence) we could argue that $\sigma+T\vdash\neg\phi$, for some sentence $\sigma\in S$. But this, together with $T+\neg\phi\vdash\sigma$ would imply that $T\vdash\sigma\longleftrightarrow \neg\phi$, contrary to the premiss of our argument. 


We have shown that $S+T\nvdash\neg\phi$. Therefore there is a structure $k$ satisfying $S+T+\phi$. Evidently $k$ is a field. Recalling that $S$ is the set of universal consequences of $T+\neg\phi$, we can now use the Lemma to conclude that $k$ extends to a field $K$ satisfying $\neg\phi$. This proves that $\phi$ is not preserved under extensions. 


I'd love some feedback on this!


