field axioms
The field axioms are the short list of rules that say arithmetic behaves the way you learned in school. They guarantee that addition and multiplication are commutative and associative, that there are special numbers zero and one, that every number has a negative, that every nonzero number has a reciprocal, and that multiplication distributes over addition.
Spelled out, a field is a set with two operations, addition and multiplication, satisfying: both operations are associative and commutative; addition has an identity 0 and each element a has an additive inverse minus a; multiplication has an identity 1 different from 0 and each nonzero element has a multiplicative inverse; and the distributive law a times the sum of b and c equals a times b plus a times c holds. These nine or so rules are all you may use when proving algebraic facts from scratch.
The rationals, the reals, and the complex numbers are all fields. Crucially, the field axioms say nothing about order or about limits, so they alone do not capture what is special about the reals. They are the algebraic foundation onto which order and completeness are later added.
From the field axioms alone you can prove little facts like a times 0 equals 0 and minus a times minus b equals a times b. They are theorems, not extra axioms.