Page 247 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes (v) Identity Elements: The set F contains an additive identity element, denoted by 0, such that
for all a in F,
a + 0 = a and 0 + a = a.
The set F also contains a multiplicative identity element, denoted by 1 (and assumed to be
different from 0) such that for all a in F,
a· 1 = a and 1· a = a.
(vi) Inverse Elements: For each a in F, the equations
a + x = 0 and x + a = 0

have a solution x in F, called an additive inverse of a, and denoted by -a. For each nonzero
element a in F, the equations
a· x = 1 and x· a = 1

have a solution x in F, called a multiplicative inverse of a, and denoted by a .
-1
Definition: Let F be a field. For a , a , . . . , a , a in F, an expression of the form
1
0
m
m-1
a x + a x + · · · + a x + a 0
m-1
m
m
1
m-1
is called a polynomial over F in the indeterminate x with coefficients a , a , . . . , a . The set of
m-1
0
m
all polynomials with coefficients in F is denoted by F[x]. If n is the largest nonnegative integer
such that a  0, then we say that the polynomial
n
f(x) = a x + · · · + a 0
n
n
has degree n, written deg(f(x)) = n, and a is called the leading coefficient of f(x). If the leading
n
coefficient is 1, then f(x) is said to be monic.
Two polynomials are equal by definition if they have the same degree and all corresponding
coefficients are equal. It is important to distinguish between the polynomial f(x) as an element
of F[x] and the corresponding polynomial function from F into F defined by substituting elements
of F in place of x. If f(x) = a x + · · · + a and c is an element of F, then f(c) = a c + · · · + a . In fact,
m
m
0
m
0
m
if F is a finite field, it is possible to have two different polynomials that define the same polynomial
function. For example, let F be the field Z and consider the polynomials x - 2x + 1 and 4x + 1. For
5
5
any c in Z , by Fermat’s theorem we have c  c (mod 5), and so
5
5
c - 2c + 1  - c + 1  4c + 1 (mod 5),
5
which shows that x - 2x + 1 and 4x + 1 are identical, as functions.
5
For the polynomials
f(x) = a x + a x + · · · + a x + a 0
m
m-1
m-1
1
m
and
g(x) = b x + b x + · · · + b x + b ,
n
n-1
1
0
n
n-1
the sum of f(x) and g(x) is defined by just adding corresponding coefficients. The product f(x)g(x)
is defined to be
a b x n+m + · · · + (a b + a b + a b )x + (a b + a b )x + a b .
2
0 0
m n
0 1
1 0
0 2
1 1
2 0
The coefficient c of x in f(x)g(x) can be described by the formula
k
k
c = k i 0a i b k i .
k 


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