TITLE: Formulas in a Field and in a Field Extension
QUESTION [8 upvotes]: 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.

REPLY [4 votes]: This was intended to be a comment to SJR's post, but was too long.
Here is a proof that  any formula that is  preserved downwards in models of $T$ is  equivalent in $T$ to a universal formula, from which we deduce that formulas preserved upwards are equivalent to existential formulas.
Let $T$ be an $\cal L$-theory.  Suppose $\phi(\bar v)$ is preserved downwards in models of $T$.
Let $\Gamma(\bar v)=\{\psi(\bar v): \psi$ is universal and $T\models \phi(\bar v)\rightarrow \psi(\bar v)\}$.
We claim that $T+\Gamma(\bar v)\models \psi(\bar v)$.  If that happens then there are $\psi_1,\dots,\psi_n\in\Gamma(\bar v)$ such that $T\models \phi(\bar v)\leftrightarrow (\psi_1(\bar v)\land\dots\land\psi_n(\bar v)$.
Suppose not.  Then there is ${\cal M}\models T$ and $\bar a\in \cal M$ such that ${\cal M}\models \psi(\bar a)$
for $\psi\in\Gamma$ and ${\cal M}\models \neg\phi(\bar a)$.
Let $T_1= T+ \phi(\bar a) + $ atomic diagram of $\cal M$.  If $T_1$ is satisfiable there is ${\cal N}\models T$
with ${\cal M}\subset {\cal  N}$ and ${\cal N}\models \phi(\bar a)$, contradicting the fact that $\phi$ is preserved downward.  Thus $T_1$ must be unsatisfiable.  Thus there is a quantifier free formula $\theta(\bar v,\bar w)$ 
and $\bar b\in \cal M$ such that $$T\models \theta(\bar a,\bar b)\rightarrow \neg\phi(\bar a).$$ Since the parameters $\bar a$ and $\bar b$ don't occur in $T$,
$$T\models \forall \bar v\ (\phi(\bar v)\rightarrow\forall \bar w\ \neg\theta(\bar v,\bar w))$$ 
But then $\forall \bar w\ \neg\theta(\bar v,\bar w)\in \Gamma$ contradicting the fact that ${\cal M}\models
\theta(\bar a,\bar b)$.