Elements of Geometric Stability Theory
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چکیده
1 Completeness and quantifier elimination for some classical theories We first work out a basic example, with a proof that demonstrates geometro-algebraic, as opposed to syntactical, methods in model theory. We denote ACF p the theory of algebraically closed fields of characteristic p. Theorem 1.1 ACF p is complete and allows quantifier elimination in the language (+, ·, 0, 1) First we prove Lemma 1.0.1 (weak form of Steinitz' Theorem) Let B and C of the same uncountable cardinality µ be in ACF p. Then any isomorphism α 0 : B 0 → C 0 between subfields B 0 ⊆ B and C 0 ⊆ C of cardinality less than µ can be extended to an isomorphism α : B → C. Proof We enumerate the fields and proceed back-and-forth constructing α i : B i → C i of cardinality less than µ. Suppose B i and C i are isomorphic and of cardinality less than µ. Take the first b ∈ B not in B i. If b is transcendental over B i then by cardinality considerations we can find c transcendental over C i. Then B i (b) ∼ = B i (x) ∼ = C i (x) ∼ = C i (c).
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