Page 221 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes where n is a non-negative integer and a , a,, ..., a  R.
0 n
While discussing polynomials we will observe the following conventions. We will

(i) write x as 1, so that we will write a for a x ,
0
0
0
0
(ii) write x as x,
1
(iii) write x instead of 1 .x (i.e., when a = l),
m
m
m
(iv) omit terms of the type O.x .
m
Thus, the polynomial 2 + 3x – 1.x is 2x + 0.x + 3x + (–1)x .
3
2
3
2
1
0
Henceforth, whenever we use the word polynomial, we will mean a polynomial in the
n
indeterminate x. We will also be using the shorter notation  a x for the polynomial
i
i
i 0

a + a x+ ... + a x .
n
n
1
0
Let us consider a few mox basic definitions related to a polynomial.
Definition: Let a + a, x + ... + a, x be a polynomial over a ring R. Each of a ,a , . . ., a, is a coefficient
n
0
l
n
of this polynomial. If a,  0, we call a, the leading coefficient of this polynomial.
If a = 0 = a = ... = a , we get the constant polynomial, a . Thus, every element of R is a constant
2
1
n
0
polynomial.
In particular, the constant polynomial 0 is the zero polynomial.
It has no leading coefficient.
Now, there is a natural way of associating a non-negative integer with any non-zero polynomial.
Definition: Let a,, + a, x + . . . + a,, x be a polynomial over a ring R, where a,  0. Then we call the
n
integer n the degree of this polynomial, and we write
n
deg   a x i    n,of a,,  0.
i
 i 0 

We define the degree of the zero polynomial to be – . Thus, deg 0 = –.
Let us consider some examples.
(i) 3x + 4x + 5 is a polynomial of degree 2, whose coefficients belong to the ring of integers
2
Z. Its leading coefficient is 3.
(ii) x + 2x + 6x + 8 is a polynomial of degree 4, with coefficients in Z and leading coefficient
4
2
2. (Note that this polynomial can be rewritten as 8 + 6x + x + 2x .)
2
4
(iii) Let R be a ring and r  R, r  0. Then r is a polynomial of degree 0, with leading coefficient
r.
Before giving more examples we would like to set up some notation.
Notation: We will denote the set of all polynomials over a ring R by R[x]. (Please note the use of
the square brackets [ ]. Do not use any other kind of brackets because R [x] and R (x) denote
different sets.)
 n 
Z .
i
i
Thus, R[x] =  a x a  R   0, 1,...n, where n  0, n  
1
i
 i 0 

We will also often denote a polynomial a + a x + . . . + a x by f(x), p (x), q(x), etc.
n
0 1 n
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