The Lefschetz principle was formulated and illustrated the first time in:

S. Lefschetz, *Algebraic Geometry*, Princeton University Press, 1953.

The basic idea is that every equation over some algebraically closed field of characteristic $0$ only involves finitely many elements, which generate a subfield isomorphic to a subfield of $\mathbb{C}$. But as Seidenberg points out in

A. Seidenberg, *Comments on Lefschetz's principle*, American American Monthly (65), No. 9, Nov. 1958, 685 - 690

Lefschetz has not given a rigorous proof and it is not clear at all if it holds when analytical methods over $\mathbb{C}$ are used. Tarski's classical result that the theory of algebraically closed fields of characteristic $0$ admits quantifier elimination and therefore all models are elementary equivalent is called the "Minor Lefschetz principle", because it does not apply to prominent examples such as Hilbert's Nullstellensatz.

A precise formulation, with a short proof, which works in *every* characteristic, can be found here:

Paul C. Eklof, *Lefschetz's Principle and Local Functors*, Proc. AMS (37), Nr. 2, Feb. 1973, online

In the language of that paper, the principle states the following: Let $F$ be a functor from universal domains of characteristic $p$ ( = algebraically closed field of infinite transcendence degree over $\mathbb{F}_p$) to some category of many-sorted structures with embeddings, which satisfies the following finiteness condition: If $K \subseteq L$ is an extension, then every finite subset of $F(L)$ is already included in the image of a subextension of finite transcendence degree over $K$.

Then, for all $K,L$, we have that $F(K)$ and $F(L)$ are elementary equivalent.

For a specific statement one wants to prove using the Lefschetz princple, one can take $F(K)$ to be the collection of all "relevant algebraic geometry over $K$".

A generalization is treated in:

Gerhard Frey, Hans-Georg Rück, *The strong Lefschetz Principle in Algebraic Geometry*, manuscripta math. (55), 385 - 401 (1986)

isa precise theorem which one might quote (the one by Eklof)? In practice one then has to check only 1), whereas 2) (and 3), namely the reason why this does the job) may be omitted. $\endgroup$1more comment