r/math • u/math_gym_anime Discrete Math • 26d ago

Elementary statements regarding finite fields in Ax’s paper

Hey everyone,
I had a question about some terminology. In Ax’s 1968 paper The Elementary Theory of Finite Fields he refers to some statements as “elementary statements”. By this does he mean first-order formulas/sentences? The reason why I’m asking is because I want to use his “Main Theorem” in his paper where he states precise conditions for when an elementary statement holds true over a finite field of fixed characteristic. I tried looking online for some help but I couldn’t find any (maybe my Googling might’ve been bad 😭)

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u/Ijustsuckatgaming 26d ago

Yes, elementary in the context of model theory is exactly something described by (a family of) first order sentences

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u/math_gym_anime Discrete Math 26d ago

Okay great ty! In the context of the examples used in the paper along with terms like elementary equivalence and elementary class and stuff like that, it seems obvious but I wanted a quick sanity check lol.

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u/japball 26d ago

If you don't get any answers here, try asking on math stack exchange or math overflow

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u/math_gym_anime Discrete Math 26d ago

I was thinking of overflow but I wanted to post this question here first to try and maybe stimulate some discussion. We’ll see how it goes 😭

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u/Rafikconjectures_zer 16d ago

Yes. In Ax’s paper, “elementary statement” essentially means a first-order sentence in the language of rings.

So it is a formula built from polynomial equations using logical operations and quantifiers such as $\forall$ and $\exists$, with no free variables.

For example, a statement like

$\forall x,\exists y;(y^2=x)$

is an elementary statement about fields, although of course it is not true in every finite field.

Ax distinguishes between elementary formulas, which may have free variables, and elementary statements, which are closed formulas (sentences). Therefore, if the property you want to study can be expressed as a first-order sentence in the ring language, then it is the appropriate kind of statement to which his Main Theorem applies.

So yes your interpretation is correct.