Lindenbaum's lemma and algebraic closures Lindenbaum's lemma in predicate logic states that every consistent, first order theory $K$ has an extension to a consistent, complete extension.  This sounds very similar to the theorem  that every field has an algebraic closure.  In fact, the process of building such a theory is similar to building a field extension (i.e.  Given a consistent theory $K$ such that it's not the case that $\vdash_K \neg \beta$, let $K'$ be the theory obtained from $K$ by introducing $\beta$ as an axiom).   The complete, consistent extension is obtained inductively in the manner.   
Are there any deeper connections between Lindenbaum's lemma and algebraic closures?
 A: I am not aware of any intrinsic (non-formal) similarity between the two theorems. So I think it's just the very similar form of proofs they admit.
This begins with the somewhat similar definitions ("a structure that contains some type of witness for every element of some set") of the sought-after structures:


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*A first-order theory $K$ is complete if it decides every formula $\phi$, i.e. for all $\phi$, either $K \vdash \phi$ or $K \vdash \neg \phi$.

*A field $F$ is algebraically closed if each polynomial $p \in F[X]$ has a root in $F$.


This similarity is reflected by the following proofs by means of Zorn's Lemma (details omitted):


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*Given a consistent theory $(K_0,\Phi_0)$ ($\Phi_0$ is the set $\{\phi: K_0\vdash \phi\text{ or }K_0\vdash \neg\phi\}$). Let $(P, \le)$ be the poset with consistent theories $(K,\Phi)$ extending $(K_0,\Phi_0)$ as elements. Let $(K,\Phi) \le (K',\Phi')$ iff $\Phi \subseteq \Phi'$. It is non-empty and by the compactness theorem, $(\bigcup_i K_i, \bigcup_i \Phi_i)$ is an upper bound for a chain $(K_i,\Phi_i)_i$. By Zorn's Lemma, we have a maximal element, which is a complete and consistent theory extending $K_0$.

*Given a field $(F_0,Z_0)$ ($Z_0$ is the set of polynomials $\{p\in F_0[X]: \exists f \in F_0\,p(f)=0\}$). Let $(P,\le)$ be the poset with fields $(F,Z)$ containing $(F_0,Z_0)$ as elements (NB. I gloss over the matter of selecting a set of fields, which would obscure the main idea at work). Let $(F,Z) \le (F',Z')$ iff $Z \subseteq Z'$. Then $(\bigcup_i F_i,\bigcup_i Z_i)$ is an upper bound for a chain $(F_i,Z_i)_i$. By Zorn's Lemma, we have a maximal element, which is an algebraically closed extension of $F$.


Besides these rather similar proofs (which rely on some finiteness property of the relevant elements: the compactness theorem cq. that a polynomial has finitely many nonzero coefficients) I don't see any connection.
