13.1 Model theory of fields

Model theory of fields explores the mathematical structures of fields using first-order logic. This topic connects algebraic properties with logical formalism, examining how field axioms and properties can be expressed and analyzed within the framework of model theory.

In this section, we dive into fundamental concepts like algebraic closure, field extensions, and the axioms of field theory. We also explore the completeness and categoricity of algebraically closed fields, as well as types and definable sets in field theory.

Fields in Model Theory

Fundamental Concepts of Fields

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• Fields serve as mathematical structures with two binary operations (addition and multiplication) satisfying specific axioms
  • Axioms include commutativity, associativity, distributivity, and existence of identity and inverse elements for both operations
• Model theory examines fields as first-order structures in the language of rings
  • Language typically denoted as L = {+, ·, 0, 1}
  • • and · represent binary function symbols
  • 0 and 1 represent constant symbols
• Field characteristic determines the smallest positive integer n where n · 1 = 0, or 0 if no such n exists
  • Characteristic 0 fields encompass rational, real, and complex numbers
  • Prime characteristic fields include finite fields (Galois fields)

Algebraic Closure and Extensions

• Algebraic closure emerges as a fundamental concept in field theory
  • Algebraically closed field contains a root for every non-constant polynomial
• Field theory in model theory concentrates on properties expressible in first-order logic
  • Focuses on existence of solutions to polynomial equations
• Subfields and field extensions play crucial roles in model theory of fields
  • Emphasizes algebraic extensions (elements satisfying polynomial equations)
  • Explores transcendental extensions (elements not satisfying any polynomial equation)

Axioms of Field Theory

Core Field Axioms

• Field theory incorporates axioms for fundamental algebraic properties
  • Associativity: $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$ for all elements a, b, c
  • Commutativity: $a + b = b + a$ and $ab = ba$ for all elements a, b
  • Distributivity: $a(b + c) = ab + ac$ for all elements a, b, c
• Axioms establish existence of additive and multiplicative identities and inverses
  • Additive identity: $a + 0 = a$ for all elements a
  • Multiplicative identity: $a · 1 = a$ for all elements a
  • Additive inverse: For every a, there exists -a such that $a + (-a) = 0$
  • Multiplicative inverse: For every non-zero a, there exists a^(-1) such that $a · a^(-1) = 1$
• Crucial axiom $0 ≠ 1$ distinguishes fields from trivial ring with one element

Consequences and Derived Properties

• Field axioms lead to important consequences and derived properties
  • Uniqueness of additive and multiplicative identities and inverses
  • Division by any non-zero element becomes possible
• No zero divisors property emerges as a consequence of field axioms
  • If $xy = 0$, then $x = 0$ or $y = 0$ (not needed as a separate axiom)
• Theory of fields requires infinitely many axioms to exclude fields of each finite characteristic
  • Not finitely axiomatizable
• Specific characteristic theories (0 or prime p) can be finitely axiomatized
  • Add appropriate characteristic axiom to the general field axioms
• Basic algebraic identities derive from field axioms
  • $(-1)(-1) = 1$
  • $(-a)b = -(ab)$ for all elements a and b

Completeness and Categoricity of Algebraically Closed Fields

Completeness of Algebraically Closed Fields

• Theory of algebraically closed fields (ACF) exhibits completeness
  • For any sentence in the language of rings, either the sentence or its negation proves from ACF
• Completeness of ACF demonstrates through quantifier elimination
  • Every formula in the language of rings equates to a quantifier-free formula in ACF
• ACF admits elimination of imaginaries
  • Simplifies study of definable sets and types in ACF models

Categoricity in Algebraically Closed Fields

• ACF lacks categoricity in any infinite cardinality
  • Non-isomorphic models exist for each infinite cardinality
• Theory of algebraically closed fields of fixed characteristic shows completeness and categoricity
  • ACF_0 for characteristic 0
  • ACF_p for prime characteristic p
  • Categorical in uncountable cardinalities
• Categoricity in uncountable cardinalities for ACF_0 and ACF_p results from two factors
  • Löwenheim-Skolem theorem
  • Isomorphism between any two algebraically closed fields of same uncountable cardinality and characteristic

Types and Definable Sets in Field Theory

Types in Field Theory

• Types in field theory correspond to prime ideals in the polynomial ring over the field
  • Maximal ideals correspond to complete types
• Every type in ACF exhibits stationarity
  • Possesses unique non-forking extension to any superset of its domain
• Strongly minimal formulas in ACF (e.g., "x = x") have geometric dimension theory
  • Corresponds to transcendence degree in field theory

Definable Sets and Closures

• Definable sets in field theory comprise Boolean combinations of polynomial zero sets
  • Also known as constructible sets in algebraic geometry
• Tarski-Seidenberg theorem states projectability of definable sets in ACF
  • Projection of a definable set in ACF remains definable
  • Equivalent to quantifier elimination in ACF
• Algebraic closure of subset A in ACF model represents smallest definably closed set containing A
  • Coincides with model-theoretic algebraic closure
• Definable closure of set A in ACF model equates to field-theoretic algebraic closure of A within the model