- 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
*F*^{2}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*).

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 (iv) For allF,0,-,+) is agroup. Familiar consequences are the right and leftcancellation rules: if, for anya,b,cinF, we havea+c=b+corc+a=b+a, thena=b. This is proved by adding (-c) to both sides from the right or left respectively. In particular,a+a=aif and only ifa=0. Likewise, for anya,binF, the equationa+x=bhas the unique solutionx=b+(-a), usually abbreviatedx=b-a(do not confuse this binary operation of ``subtraction'' with the unary additive inverse!). Another standard consequence of (iii) is that, for anya_{1},a_{2}, ... ,ain_{n}F, the suma_{1}+a_{2}+ ... +ais the same no matter how it is parenthesized. (In how many ways_{n}canthat expression be parenthesized?)

Conditions (i), (ii), (iii) assert that (v) For allF,0,-,+) is acommutative group, a.k.a.abelian grouporadditive 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.

vi) For all

vii) For all

In particular, restricting (v), (vi) and (vii) toviii) For allF, we are asserting that (^{*}F,1,^{*}^{-1},*) is a group.

So, the group (ix) For allF,1,^{*}^{-1},*) is also abelian. IfFsatisfies all the field axioms except (viii), it is called askew field; the most famous example is thequaternionsof 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.

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

For allainF,a*0=0*a=0.

Thus also:

For alla,binF,ab=0 if and only ifa=0 orb=0 (or both).

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

- 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].