TITLE: Defining the existence of a non algebraic element in the language $L:= \{0,1,+,\cdot\}$
QUESTION [7 upvotes]: I raise following question after reading this post.
Is it possible in the language $L:= \{0,1,+,\cdot\}$ to write sentences for which a model will necessarily contain a copy of $\mathbb Q$ and a non algebraic element?
If yes, what is this set of sentences? If not can you provide a proof?
Thanks.
I added the word necessarily to precise the question.

REPLY [2 votes]: Here is another example, similar to what Alex gave.
Let $T$ be the theory of fields asserting that every nonconstant polynomial with coefficients in $\mathbb{Z}$ has a root, that the charactersitic is zero, and that not every element has a square root. This theory is finitely satisfiable: simply pick a finite field of large characteristic containing the roots of any given collection of polynomials with coefficients in $\mathbb{Z}$, making sure that the characteristic is also larger than the absolute values of these coefficients.
Every model of this theory will contain $\mathbb{Q}$, since it will be a field of characteristic zero. It will also contain the algebraic clousre of $\mathbb{Q}$, since every polynomial with integer coefficients has a root. However, the field must be bigger than that, since not every element has a square root.