Page 246 - DMTH403_ABSTRACT_ALGEBRA

Unit 25: Roots of a Polynomial

Hence, the theorem is true for all n  1. Notes
Using this result we know that, for example, x –1  Q[x] can’t have more than 3 roots in Q.
3
In Theorem 1 we have not spoken about the roots being distinct. But an obvious corollary of
Theorem 1 is that
if f(x)  F[x] is of degree n, then f(x) has st most n distinct roots in F.

We will use this result to prove the following useful theorem.
Theorem 2: Let f(x) and g(x) be two non-zero polynomials of degree n over the field F. If there
exist n+l distinct elements a,,.. .,a , in F such that f(a ) = g(a )  i = I , ..., n+l, then f(x) = g(x).
i
i
n+1
Proof: Consider the polynomial h(x) = f(x) = g(x)
Then deg h(x)  n, but it has n + l distinct roots a,, ..., a .
n+1
This is impossible, unless h(x) = 0, i.e., f(x) = g(x).

3
Example: Prove that x  5x  Z [x] has more roots than its degree. (Note that Z is not
6
6
a field.)
Solution: Since the ring is finite, it is easy for us to run through all its elements and check which
of them, are roots of
3
f(x)  x  5x.
So, by substitution we find that

f(0) = 0 = f(1) = f(2) = f(3) = f(4) = f(5).
In fact, every element of Z is a zero of f(x). Thus, f(x) has 6 zeros, while deg f(x) = 3.
6
So far, we have been saying that a polynomial of degree n over F has at most n roots in Fa. It can
happen that the polynomial has no root in F. For example, consider the polynomial x + 1  R[x].
2
You know that it can have 2 roots in R, at the most. But as you know, this has no roots in R (it has
two roots, i and –i, in C).
We can find many other examples of such polynomials in R[x]. We call such polynomials
irreducible over R. We shall discuss them in detail in the next units.
Now let us end this unit by seeing what we have covered in it.
Definition: Let F be a set on which two binary operations are defined, called addition and
multiplication, and denoted by + and · respectively. Then F is called a field with respect to these
operations if the following properties hold:
.
(i) Closure: For all a,b in F the sum a + b and the product a b are uniquely defined and belong
to F.
(ii) Associative Laws: For all a,b,c in F,
a + (b + c) = (a + b) + c and a· (b· c) = (a· b)· c.

(iii) Commutative Laws: For all a,b in F,
a + b = b + a and a · b = b· a.
(iv) Distributive Laws: For all a, b, c in F,
a· (b + c) = (a· b) + (a· c) and (a + b)· c = (a· c) + (b· c).

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