Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

UNIT-IV 99
x 3 = (0, 0, 0, 1,…)
. . . . . . . .. . . . . . .
. . . . . . . .. . . . . . .
x n = ( 0
1
,0,0,....,0 1, 0,….)
4 2 43
n terms
Therefore for (a, 0, 0, ….) ∈ Q we have
(a, 0, 0,…) x = (0, a, 0, …..)
(a, 0, 0, …) x 2 = (0, 0, a, ….)
. . . . . . . .. . . . . . . . . . . . . . .
(a, 0, 0, ….) x n = ( 0
1
,0,0,....,0
4 2 43
, a, 0, ….) (ii)
n terms
If (a 0 , a 1 , .., a n , 0, …) be any arbitrary element of the polynomial ring P, then by (ii) we have
(a 0 , a 1 , a 2 ,…., a n , 0, …) = (a 0 , 0, ….) + (0, a 1 ,…) + … + ( 0
1
,0,0,....,0
4 2 43
, a n , 0, ….)
n terms
= (a 0 , 0, …) + (a 1 , 0, ….) (0, 1, 0,…) + ….
+ (a n , 0, 0,…) ( ( 0,0,0,...,0,0, 1, 0,…)
1 4 2 4 43
n terms
= (a 0 , 0, …) + (a 1 , 0, 0, …) x + … + (a n , 0, 0,…) x n
= a 0 + a 1 x + a n x n (by (i))
Hence every element (a 0 , a 1 , a 2 ,…) of P can be denoted by
a 0 + a 1 x + a 2 x 2 + … a n x n .
* The numbers a 0 , a 1 ,…, a n are called coefficients of the polynomial. If the coefficient a n of x n is nonzero,
then it is called leading coefficient of a 0 + a 1 x + … + a n x n .
* A polynomial consisting of only one term a 0 is called constant polynomial.
Example. If R is a commutative ring with unity, prove that R[x] is also a commutative ring with unity.
Degree of Polynomial. Let f(x) = a 0 + a 1 x + … a n x n be a polynomial. If a n ≠ 0 , then n is called the
degree of f(x). We denote it by deg f(x) = n.
It is clear that degree of a constant polynomial is zero.
If
and
f(x) = a 0 + a 1 x + a 2 x 2 + … a m x m , a m ≠ 0
g(x) = b 0 + b 1 x + … b n x n , b n ≠ 0
are two elements of R[x], then
deg f(x) = m and deg g(x) = n and
f(x) + g(x) = (a 0 + a 1 x + a 2 x 2 + … + a m x m ) + (b 0 +b 1 x+b 2 x 2 + … + b n x n )
If m = n and a m + b n ≠ 0, then
f(x) + g(x) = (a 0 + b 0 ) + (a 1 +b 1 )x + … + (a m + b m )x m
Therefore in this case
deg [f(x) + g(x)] = m.
It is also clear that if m = n and a m + b m = 0 , then
deg [f(x) + g(x)] < m.