Field axioms

axiom 1 Closure: the combination (hereafter indicated by addition or multiplication) of any two elements in the set produces an element in the set.

axiom 2 Addition is commutative: a + b = b + a for any elements in the set.

axiom 3 Addition is associative: a + (b + c) = (a + b) + c for any elements in the set.

axiom 4 Additive identity: there exists an element 0 such that a + 0 = a for every element in the set.

axiom 5 Additive inverse: for each element a in the set, there exists an element -a such that a + (-a) = 0.

axiom 6 Multiplication is associative: a(bc) = (ab)c for any elements in the set.

axiom 7 Distributive law: a(b + c) = ab + ac and (a + b)c = ac + bc for any elements in the set.

axiom 8 Multiplication is commutative: ab = ba for any elements in the set.

axiom 9 Multiplicative identity: there exists an element 1 such that 1a = a for any element in the set.

axiom 10 Multiplicative inverse: for each element a in the set, there exists an element a^-1 such that aa^-1 = 1.