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The signature of (unital) rings has 2 constants 0 and 1, two binary functions + and ×, and, optionally, one unary inverse functions − −1. Rings Axioms: Addition makes the ring into an abelian group, multiplication is associative and has an identity 1, and multiplication is left and right distributive. Commutative rings The axioms for rings plus ∀x ∀y xy=yx. Fields The axioms for commutative rings plus ∀x ∃y xy=1 and ¬ 1=0. Many of the examples given here have only universal, or algebraic axioms. The class of structures satisfying such a theory has the property of being closed under substructure. For example, a subset of a group closed under the group actions of multiplication and inverse is again a group. Since the signature of fields does not usually include multiplicative and additive inverse, the axioms for inverses are not universal, and therefore a substructure of a field closed under addition and multiplication is not always a field. This can be remedied by adding unary inverse functions to the language. For any positive integer n the property that all equations of degree n have a root can be expressed by a single first-order sentence: ∀ a1 ∀ a2... ∀ an ∃x (...((x+a1)x +a2)x+...)x+an = 0 Perfect fields The axioms for fields, plus axioms for each prime number p stating that if p 1 = 0 (i.e. the field has characteristic p), then every field element has a pth root. Algebraically closed fields of characteristic p The axioms for fields, plus for every positive n the axiom that all polynomials of degree n have a root, plus axioms fixing the characteristic. The classical examples of complete theories. Categorical in all uncountable cardinals. The theory ACFp has a universal domain property, in the sense that every structure N satisfying the universal axioms of ACFp is a substructure of a sufficiently large algebraically closed field M \models ACF_0 , and additionally any two such embeddings N → M induce an automorphism of M. Finite fields. The theory of finite fields is the set of all first-order statements that are true in all finite fields. Significant examples of such statements can, for example, be given by applying the Chevalley–Warning theorem, over the prime fields. The name is a little misleading as the theory has plenty of infinite models. Ax proved that the theory is decidable. Formally real fields These are fields with the axiom For every positive n, the axiom ∀ a1 ∀ a2... ∀ an a1a1+a2a2+ ...+anan=0 → a1=0∨a2=0∨ ... ∨an=0 (0 is not a non-trivial sum of squares). Real closed fields Axioms: ∀x ∃y x=yy ∨ x+yy=0. For every odd positive n, the axiom stating that every polynomial of degree n has a root. For every positive n, the axiom ∀ a1 ∀ a2... ∀ an a1a1+a2a2+ ...+anan=0 → a1=0∨a2=0∨ ... ∨an=0 (0 is not a non-trivial sum of squares). The theory of real closed fields is effective and complete and therefore decidable (Tarski). p-adic fields: Ax & Kochen (1965) showed that the theory of p-adic fields is decidable and gave a set of axioms for it. |
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