Let $L$ be a first-order language and $M$ be an $L$-structure. Let $D \subseteq M^n$ . Let us say $D$ is definable in $M$ if for some finite set (possibly empty) $A=\{a_1,...,a_m\} \subseteq M$ and some formula $\psi[x_1,...,x_n,y_1,...,y_m]$ , $D=\{(b_1,...,b_n)\in M^n : M\vDash \psi[b_1,...,b_n,a_1,...,a_m] \}$.

(i.e. $D$ is definable by a finite set of parameters according to this definition https://en.wikipedia.org/wiki/Definable_set ).

Now consider the first order theory of commutative rings. Take $M=\mathbb C[[X]]$ (the formal power series ring with complex coefficients) .

My questions are : Is $\mathbb C$ definable in $\mathbb C[[X]]$ ? Is $\mathbb C[X]$ definable in $\mathbb C[[X]]$

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