Could someone elaborate me what the sentence "The elementary theory of finite fields is decidable" means? I'm not sure that for example if I take $x\in \mathbb{F}_4$ and $y\in \mathbb{F}_5$ then can I form a first order predicate logic sentence that contains both $x$ and $y$? And what kind of inputs we can make to a Turing machine if we have elements from many fields?
2026-04-06 22:42:45.1775515365
Elementary theory of an algebraic structures
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This means that there is a Turing machine which decides whether or not a statement in the language of fields is provable from the theory of finite fields. If we consider the theory as closed under consequences, then it is the same as asking whether or not the statement is in the theory. The Turing machine takes a statement in the language, not a statement about models of the theory.
See also Wikipedia.
Note that theories live in the syntax, whereas $\Bbb F_4$ and $\Bbb F_5$ are semantics. You can't write a statement about different structures in first-order logic.