Page 226 - DMTH403_ABSTRACT_ALGEBRA

Unit 22: Polynomial Rings

Since a , a , ... and b , b . .. . are all zero, Notes
n+1 n+2 m+1 m+2
c m+n = a b
n m
Now, if R is without zero divisors, then a b  0, since a,  0
n m
and b  0. Thus, in this case,
deg (f(x) g (x)) = deg f(x) + deg g (x).
On the other hand, if R has zero divisors, it can happen that a,b, = 0. In this case,

deg (f (x) g (x)) < m+n = deg f(x) + deg g(x).
Thus, our theorem is proved.
The following result follows immediately from Theorem 3.
Theorem 4: R [x] is an integral domain <=>. R is an integral domain.

Proof: From Theorem 2, we know that R is a commutative ring with identity iff R[x] is a
commutative ring with identity. Thus, to prove this theorem we need to prove that. R is without
zero divisors iff R [x] is without zero divisors.
So let us first assume that R is without zero divisors.
Let p(x) = a + a x+ ... + a x , and q(x) = b + b x +... +b x
0
m m
l
1
0
n n
be in R [x], where a,  0 and b,  0.
Then, in Theorem 3 we have seen that deg .(p (x) q (x)) = m + n  0.
Thus, p (x) q (x)  0
Thus, R [x] is without zero divisors.
Conversely, let us assume that R [x] is without zero divisors. Let a and b be non-zero elements
of R. Then they are non-zero elements of R [x] also. Therefore, ab  0. Thus, R is without zero
divisors. So, we have proved the theorem.

Now, you have seen that many properties of the ring R carry over to R’[x]. Thus, if F is a field, we
should expect F[x] to be a field also, But this is not so. F[x] can never be a field.
This is because any polynomial of positive degree in F|x| does not have a multiplicative
inverse. Let us see why.
Let f (x)  F [x] and deg f (x) = n > 0. Suppose g (x)  F [x] such that f (x) g (x) = 1. Then
0 = deg 1 = deg (f(x) g (x)) = deg f(x) + deg g (x), since F [x] is a domain.

= n + deg g (x)  n > 0.
We reach a contradiction.
Thus, F [x] cannot be a field.
But there are several very interesting properties of F [x], which are similar to those of Z, the set
of integers. In the next section we shall discuss the properties of division in F [x].

Self Assessment

1. A polynomial over a ring R in determinate X is an expression of the form .................
(a) a x + a x + a x + ...... a x n (b) a x + a x + a x + ...... a x n
0
2
3
0
n
2
1
3
1
2
0
1
2
n
(c) a x + a x + a x ...... a x n (d) a x + a x + a x ...... a x -n
-1
-1
-1
-3
-1
-1
-1
n
2
0
3
1
2
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