The theory of fields is an <a href="http://en.wikipedia.org/wiki/Decidability_(logic)#Some_undecidable_theories">undecidable theory</a>, and so one cannot give a computable procedure for deciding whether a given statement in the formal language of fields is true or not in all fields. This is a sense in which this theory is not fully understood. Indeed, the situation is that we have proved that we can have no computably complete understanding of the theory of fields.

I believe also that the theory of finite fields is an undecidable theory. 

Meanwhile, the theory is not what is called *essentially undecidable*, since the theory of fields has a complete decidable extension, namely, the theory of [real-closed fields](http://en.wikipedia.org/wiki/Real_closed_field), which is decidable. (There are also many other trivial extensions of the theory that are decidable, such as the theories of various specific finite fields.)