Page 249 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 5. x3 + 5x  z [x] has ............... roots than its degree.
6
(a) 2 (b) 3

(c) 1 (d) more
25.2 Summary

If f(x) and g(x) are non-zero polynomials in F[x], then f(x)g(x) is non-zero and deg(f(x)g(x))

= deg(f(x)) + deg(g(x)).
If f(x),g(x),h(x) are polynomials in F[x], and f(x) is not the zero polynomial, then f(x)g(x) =

f(x)h(x) implies g(x) = h(x).
Let f(x),g(x) be polynomials in F[x]. If f(x) = q(x)g(x) for some q(x) in F[x], then we say that

g(x) is a factor or divisor of f(x), and we write g(x) | f(x). The set of all polynomials
divisible by g(x) will be denoted by < g(x) >.

For any element c in F, and any positive integer k,

(x - c) | (x - c ).
k
k
Let f(x) be a non-zero polynomial in F[x], and let c be an element of F. Then there exists a

polynomial q(x) in F[x] such that
f(x) = q(x)(x - c) + f(c).
Moreover, if f(x) = q (x)(x - c) + k, where q (x) is in F[x] and k is in F, then q (x) = q(x) and
1
1
1
k = f(c).
Let f(x) = a x + · · + a belong to F[x]. An element c in F is called a root of the polynomial
m
 m 0
f(x) if f(c) = 0, that is, if c is a solution of the polynomial equation f(x) = 0 .
Let f(x) be a non-zero polynomial in F[x], and let c be an element of F. Then c is a root of f(x)

if and only if x-c is a factor of f(x). That is,
f(c) = 0 if and only if (x-c) | f(x).
A polynomial of degree n with coefficients in the field F has at most n distinct roots in F.

25.3 Keywords

Field: 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.

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.

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 non-zero
element a in F, the equations
a· x = 1 and x· a = 1

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