The field axioms listed below describe the basic properties of the four operations of arithmetic: ambition, distraction, uglification, and derision. [Don't blame me for this last; it's due to C.L.Dodgson, a.k.a. Lewis Carroll. Go here for the relevant quote from Alice.] Actually, only ambition and uglification -- er, addition and multiplication -- are needed, together with additive and multiplicative inverses and identities. A field, then, is a set F equipped with:
• Two distinguished elements called 0 and 1, which must be different
• A function from F to F called additive inverse and denoted by the unary minus sign (so the additive inverse of a is -a), and a function from F-{0} to F-{0} called multiplicative inverse that takes a to an element called a-1 (we shall usually write F* for F-{0})
• Two functions F2 to F called addition and multiplication; as usual we shall denote the images of (a,b) under these two functions by a + b and a*b (or a·b, or simply ab).
These must satisfy the following nine conditions, or axioms:

i) For all a in F, a+0=0+a=a [i.e., 0 is an additive identity]
ii) For all a in F, a+(-a)=(-a)+a=0 [this is what ``additive inverse'' means]
iii) For all a,b,c in F, a+(b+c)=(a+b)+c [i.e., addition is associative]

Conditions (i), (ii), (iii) assert that (F,0,-,+) is a group. Familiar consequences are the right and left cancellation rules: if, for any a,b,c in F, we have a+c=b+c or c+a=b+a, then a=b. This is proved by adding (-c) to both sides from the right or left respectively. In particular, a+a=a if and only if a=0. Likewise, for any a,b in F, the equation a+x=b has the unique solution x=b+(-a), usually abbreviated x=b-a (do not confuse this binary operation of ``subtraction'' with the unary additive inverse!). Another standard consequence of (iii) is that, for any a1, a2, ... , an in F, the sum a1 + a2 + ... + an is the same no matter how it is parenthesized. (In how many ways can that expression be parenthesized?)
iv) For all a,b in F, a+b=b+a [i.e., addition is commutative]
Conditions (i), (ii), (iii) assert that (F,0,-,+) is a commutative group, a.k.a. abelian group or additive group. The first alias is a tribute to N.H.Abel (1802-1829); the second reflects the fact that in general one only uses ``+'' for a group law when the group is commutative -- else multiplicative notation is almost always used.
v) For all a in F, a*1=1*a=a [i.e., 1 is a multiplicative identity]
vi) For all a in F*, a*a-1=a-1*a=1 [this is what ``multiplicative inverse'' means]
vii) For all a,b,c in F, a(bc)=(ab)c [i.e., multiplication is associative]
In particular, restricting (v), (vi) and (vii) to F*, we are asserting that (F*,1,-1,*) is a group.
viii) For all a,b in F, ab=ba [i.e., multiplication is commutative]
So, the group (F*,1,-1,*) is also abelian. If F satisfies all the field axioms except (viii), it is called a skew field; the most famous example is the quaternions of W.R.Hamilton (1805-1865). Much of linear algebra can still be done over skew fields, but we shall not pursue this in Math 55.
ix) For all a,b,c in F, a(b+c)=(ab)+(ac) and (a+b)c=(ac)+(bc). [distributive law. The second part is of course redundant by commutativity; it is required in the skew case.]

From (ix) together with the additive properties follows the basic identity:

For all a in F, a*0=0*a=0.
Proof: apply the distributive law to a(0+0) and (0+0)a, and use the fact that 0 is the only solution of x+x=x.

Thus also:

For all a,b in F, ab=0 if and only if a=0 or b=0 (or both).
Proof: If a is nonzero, multiply ab=0 by a-1 to conclude b=0.

That is, a field has no (nontrivial) zero divisors.

Note that (vi) is the only axiom using the multiplicative inverse. If we drop the existence of multiplicative inverses and axiom (vi), we obtain the structure of a ring (commutative with unity). For example, Z is a ring which is not a field. A ring may have nontrivial zero divisors (you have seen an example of this already in class); if it does not, it is called a domain.

A vector space over a field F is an additive group V (the ``vectors'') together with a function (``scalar multiplication'') taking a field element (``scalar'') and a vector to a vector, as long as this function satisfies the axioms
• 1*v=v for all v in V [so 1 remains a multiplicative identity],
• for all scalars a,b and all vectors u,v we have a(u+v)=au+av and (a+b)u=au+bu [distributive properties].
Notice that these axioms do not use the multiplicative inverse; they can thus be used equally when F is any ring, in which case the resulting structure is called a module over F. But multiplicative inverses are used to prove the basic theorems on vector spaces, so those theorems do not hold in the more general setting of modules; for instance one cannot speak of the dimension of a general module, even over Z.