As the title explains, I'm trying to work out if the axiomatisation of the first-order theory $Σ$ of ordered fields in the language $L := \lbrace+, ·; <; 0, 1\rbrace$ is maximal consistent.
It's easy to see it's consistent since it has a model (e.g. $\mathbb{R}$), but I can't think of how to tell if it's maximal or not.
Any help would be appreciated.
Suppose that $\Sigma$ is maximal. Then every ordered field (i.e., every model of $\Sigma$) satisfies exactly the same $L$-sentences as $\mathbb{R}$. Can you think of something expressible in $L$ (something like an equality or inequality of polynomials with integer coefficients) that is true in $\mathbb{R}$ but not in (for instance) $\mathbb{Q}$?