i) For all a in F, we have a+0=0+a=a
[i.e., 0 is an additive identity]
ii) For all a in F, we have a+(−a)=(−a)+a=0 [this is what “additive inverse” means]
iii) For all a,b,c in F, we have a+(b+c)=(a+b)+c [i.e., addition is associative]
Conditions (i), (ii), (iii) assert thativ) For all a,b in F, we have a+b=b+a [i.e., addition is commutative]
(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, …, anin F, the sum a1 + a2 + … + anis the same no matter how it is parenthesized. (In how many ways can that expression be parenthesized?)
Conditions (i), (ii), (iii), (iv) assert thatv) For all a in F, we have a*1=1*a=a [i.e., 1 is a multiplicative identity]
(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.
In particular, restricting (v), (vi) and (vii) to F*, we are asserting thatviii) For all a,b in F, we have ab=ba [i.e., multiplication is commutative]
(F*, 1, −1, *)is a group.
So, the groupix) For all a,b,c in F, we have 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.]
(F*, 1, −1, *)is also abelian.
From (ix) together with the additive properties follows the basic identity:
For all a in F, we have 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.
For all a,b in F, we have 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.
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 much, if at all, in Math 55.
Note that (vi) is the only axiom using the multiplicative inverse. If we drop the existence of multiplicative inverses and axiom (vi), as well as the condition 0≠1, we obtain the structure of a commutative ring with unity. For example, Z is such a ring which is not a field. A ring may have nontrivial zero divisors (we shall see an example of this in class); if it does not, it is called a domain.
If we also drop axiom (viii) from the ring axioms, we have a
ring with unity which need not be commutative. An example is
the set of the Hamilton quaternions
If F is a ring that need not be a field, the notation
F* means not
Notice that these axioms do not use the multiplicative inverse; they can thus be used equally when F is any ring (even a non-commutative ring), in which case the resulting structure is called a module over F. But multiplicative inverses are used to prove most of 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.