that are true in all finite groups (there are plenty of infinite models of this theory). It is not completely trivial to find any such statement that is not true for all groups: one example is "given two elements of order 2, either they are conjugate or there is a non-trivial element commuting with both of them". The properties of being finite, or free, or simple, or torsion are not first-order. More precisely, the first-order theory of all groups with one of these properties has models that do not have this property. Rings and fields The signature of (unital) rings has two constants 0 and 1, two binary functions + and ×, and, optionally, one unary negation function −. 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 (¬ x = 0 → ∃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