On proving the equivalence of axiomatic systems that don't have the same primitive constants Let $F_1$ be the standard axiomatic system defining the concept of an algebraic field. $F_1$ has two primitive binary operations: $+$ (addition) and $\cdot$ (multiplication).
Let $F_2$ be another axiomatic system defining the concept of an algebraic field that has, in addition to the two primitive binary operations mentioned above, also the primitive unary operations $-$ (additive inverse) and $^{-1}$ (multiplicative inverse).
Intuitively, I feel that $F_1$ and $F_2$ are equivalent in the sense that every statement that can be proved in $F_1$ can also be proved in $F_2$, and vice versa. However, how can this feeling be formalized?
One obstacle that I've met when trying to formalize the equivalence of $F_1$ and $F_2$ is that since the symbol $^{-1}$ is not used in $F_1$, it can be defined there to be an object other than the multiplicative inverse, and then there will be statements in $F_1$ that won't be provable in $F_2$ or that won't even make sense in $F_2$, as when $^{-1}$ is defined to be a field scalar rather than a function.
 A: The idea you are looking for is that of conservativity. Let $\mathcal{L}_1$ and $\mathcal{L}_2$ be first order languages with $\mathcal{L}_1 \subseteq \mathcal{L}_2$. Let $T_1$ be an $\mathcal{L}_1$ theory and $T_2$ an $\mathcal{L}_2$ theory. If, for any $\mathcal{L}_1$ statement $\phi$, $T_1 \vdash \phi \iff T_2 \vdash \phi$, then $T_2$ is conservative over $T_1$.
In this context, we can let $\mathcal{L}_1$ contain the symbols $+$ and $\times$ and $T_1$ be the theory of fields in this language. Let $\mathcal{L}_2$ be the language with $+$, $\times$, $-$ and $^{-1}$ and $T_2$ be the theory of fields in this language. It is not hard to prove that $T_1 \subseteq T_2$. It may take more work to show that if $\phi$ is any $\mathcal{L}_1$ statement such that $T_2 \vdash \phi$, then $T_1 \vdash \phi$, but there is a simple model-theoretic proof of this relying on completeness: any model of $T_1$ can be expanded to a model of $T_2$ just by interpreting the $-$ and $^{-1}$ symbols correctly, so if $T_2 \vdash \phi$, then every model $F \models T_1$ will also satisfy $\phi$, and therefore $T_1 \vdash \phi$.
