I have seen fields defined with axioms that only include 2 associative axioms, 2 commutative axioms, 2 identities, 2 inverse axioms, 1 distributive axiom, as defined in Spivak's calculus.
I've also seen an extra axiom (probably redundant) in some places, where it states that a field is closed under addition and multiplication. It's obviously intuitive why that is for R, Q and so forth, but how can we show that is true with the above axioms?
It's less obvious if I have a set S with n elements that is also a field, (not necessarily a "finite field"), if I add 1+1+ ...+1+1 and do this more than n times, I must arrive at an element within S.
This sort of "looping" behaviour is obvious when we look specifically at finite fields, since it's defined this way. (Using the equivalence class definition, $[a]_k = [a']_k$ if $a = nk + a'$ where k is prime and $n \in Z$.
However, when we just relax this to a set S with n elements that is a field, why is this necessarily closed? How do you show it with just the 9 axioms from Spivak?
"I've also seen an extra axiom (probably redundant) in some places, where it states that a field is closed under addition and multiplication. It's obviously intuitive why that is for R, Q and so forth, but how can we show that is true with the above axioms?"
Addition and multiplication are binary operations
$$+:K\times K \rightarrow K:(x,y)\mapsto x+y$$
and
$$\cdot :K\times K \rightarrow K:(x,y)\mapsto x\cdot y.$$
In this way, the field $K$ is closed under addition and multiplication. Take two elements $x,y\in K$. Then the images $x+y$ and $x\cdot y$ lie by definition (add and mult are operations) in $K$.