Page 47 - Advanced Linear Algebra

Preliminaries 31

Theorem 0.31 Any finite ring has nonzero characteristic. Any finite integral
domain has prime characteristic.
Proof. We have already seen that a finite ring has nonzero characteristic. Let -
be a finite integral domain and suppose that char²-³ ~ € . If ~ , where
Á  , then h ~ . Hence, ² h ³² h ³ ~ , implying that h ~ or
h ~ . In either case, we have a contradiction to the fact that is the smallest

positive integer such that h ~ . Hence, must be prime.
-
Notice that in any field of characteristic , we have ~ for all  - .
Thus, in ,
-
~c for all  -
This property takes a bit of getting used to and makes fields of characteristic
(
quite exceptional. As it happens, there are many important uses for fields of
characteristic . It can be shown that all finite fields have size equal to a
)

positive integral power of a prime and for each prime power , there is a

finite field of size . In fact, up to isomorphism, there is exactly one finite field

of size .

Algebras
The final algebraic structure of which we will have use is a combination of a
(
vector space and a ring. We have not yet officially defined vector spaces, but
we will do so before needing the following definition, which is placed here for
easy reference.)
Definition An algebra over a field is a nonempty set , together with
7
-
7
(
three operations, called addition denoted by b ) , multiplication denoted by
(
(
)
juxtaposition and scalar multiplication also denoted by juxtaposition , for
)
which the following properties hold:
)
1 7 is a vector space over under addition and scalar multiplication.
-
)
2 7 is a ring under addition and multiplication.
)
3 If  - and Á  7 , then
² ³ ~ ² ³ ~ ² ³
Thus, an algebra is a vector space in which we can take the product of vectors,
(
or a ring in which we can multiply each element by a scalar subject, of course,
to additional requirements as given in the definition .
)