Page 46 - Advanced Linear Algebra

30 Advanced Linear Algebra

" b # ~
Hence,
" – ² c # ³ – mod

"
and so "p ~ in { , that is, is the multiplicative inverse of .

The previous example shows that not all fields are infinite sets. In fact, finite
fields play an extremely important role in many areas of abstract and applied
mathematics.

A field is said to be algebraically closed if every nonconstant polynomial
-
over has a root in . This is equivalent to saying that every nonconstant
-
-
polynomial splits over . For example, the complex field is algebraically
d
-
closed but the real field is not. We mention without proof that every field is
s
-
contained in an algebraically closed field , called the algebraic closure of .
-
-
For example, the algebraic closure of the real field is the complex field.
The Characteristic of a Ring
9
Let be a ring with identity. If is a positive integer, then by h , we simply

mean
h ~ b Äb
’••“••”
terms
Now, it may happen that there is a positive integer for which

h ~
For instance, in { , we have h ~ ~ . On the other hand, in , the
{
equation h ~ implies ~ and so no such positive integer exists.
Notice that in any finite ring, there must exist such a positive integer , since the

members of the infinite sequence of numbers
h Á h Á h Á Ã
cannot be distinct and so h ~ h for some  , whence ² c ³ h ~ .
Definition Let be a ring with identity. The smallest positive integer for
9

9
which h ~ is called the characteristic of . If no such number exists, we
say that 9 has characteristic . The characteristic of 9 is denoted by

char²9³.
If char²9³ ~ , then for any  9 , we have
h ~ bÄb ~ ² bÄb ³ ~ h ~
’••“••”
’••“••”
terms terms