A field is an ordered triple $(F, +,\cdot)$ of a set $F$ and binary operations $+,\times$ on $F$ such that $(F,+)$ and $(F\backslash 0,\times)$ are abelian groups satisfying the distributive laws
$\forall a,b,c \in F:\begin{cases} a\times(b+c)=(a\times b)+(a\times c)\newline (a+b)\times c= (a\times c)+(b\times c) \end{cases}$.
Someone mentioned that the "interplay" between the additive structure $(F,+)$ and the multiplicative structure $(F\backslash 0,\times)$ in a field is still "not well understood". Another friend mentioned that the distributive laws fully characterize the relationship between $+,\times$ by definition, but that these laws are "not fully understood".
What is this deep understanding? What is known and what is unknown? Restrict this question to specific fields if necessary, like finite.
EDIT: I should clarify that this came up during a discussion of Wang's attacks on hash functions in cryptography.
Best Answer
The theory of fields is an undecidable theory, 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.
The theory of fields is not what is called essentially undecidable, however, since the theory of fields has a complete decidable extension, namely, the theory of real-closed fields, 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.)
Furthermore, James Ax proved that the theory of finite fields is decidable. Thus, we have a computable procedure to decide whether a given statement is true or not in all finite fields, which is surely shows a very good measure of understanding in that context. (Thanks for correction of my earlier remark by Donu Arapura.)