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**Definition 4.1.1.**
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).**(v)***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.

The set F also contains a**multiplicative identity**element, denoted by 1 (and assumed to be different from 0) such that for all a in F,a

**·**1 = a and 1**·**a = a.**(vi)***Inverse elements:*For each a in F, the equationsa+x = 0 and x+a = 0

have a solution x in F, called an

**additive inverse**of a, and denoted by -a.

For each nonzero element a in F, the equationsa

have a solution x in F, called a**·**x = 1 and x**·**a = 1**multiplicative inverse**of a, and denoted by a^{-1}.

a_{m}x^{m} +
a_{m-1}x^{m-1}
+ · · · +
a_{1}x +
a_{0}

If n is the largest nonnegative integer such that a

f(x) =
a_{n}x^{n}
+ · · · +
a_{0}

If the leading coefficient is 1, then f(x) is said to be

Two polynomials are equal by definition if they have the same degree
and all corresponding coefficients are equal.
It is important to distinguish between the polynomial f(x)
as an element of F[x] and the corresponding
** polynomial function**
from F into F defined by substituting elements of F in place of x.
If f(x) =
a_{m}x^{m}
+ · · · +
a_{0}
and c is an element of F,
then f(c) =
a_{m}c^{m}
+ · · · +
a_{0}.
In fact, if F is a finite field, it is possible to have
two different polynomials that define the same polynomial function.
For example, let F be the field **Z**_{5}
and consider the polynomials
x^{5} -2x + 1 and 4x + 1.
For any c in **Z**_{5}, by
Fermat's theorem
we have c^{5}
c (mod 5), and so

c^{5} -2c + 1
-c + 1
4c + 1 (mod 5),

For the polynomials

f(x) =
a_{m}x^{m} +
a_{m-1}x^{m-1}
+ · · · +
a_{1}x +
a_{0}

and

g(x) =
b_{n}x^{n} +
b_{n-1}x^{n-1}
+ · · · +
b_{1}x +
b_{0},

the sum of f(x) and g(x) is defined by just adding corresponding coefficients. The

a_{m}b_{n}x^{n+m}
+ · · · +
(a_{2}b_{0} +
a_{1}b_{1} +
a_{0}b_{2})x^{2}
+ (a_{1}b_{0} +
a_{0}b_{1})x +
a_{0}b_{0}.

c_{k} =
a_{i} b_{k-i}.

**Proposition 4.1.5.**
If f(x) and g(x) are nonzero polynomials in F[x],
then f(x)g(x) is nonzero and

deg(f(x)g(x)) = deg(f(x)) + deg(g(x)).

**Corollary 4.1.6.**
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).

**Definition 4.1.7.**
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) >.

**Lemma 4.1.8.**
For any element c in F, and any positive integer k,

(x - c) | (x^{k} - c^{k}).

**Theorem 4.1.9. [Remainder Theorem]**
Let f(x) be a nonzero 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

**Definition 4.1.10.**
Let f(x) =
a_{m}x^{m}
+ · · · +
a_{0}
belong to F[x].
An element c in F is called a
** root**
of the polynomial f(x) if f(c) = 0, that is,
if c is a solution of the polynomial equation f(x) = 0 .

**Corollary 4.1.11.**
Let f(x) be a nonzero 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).

**Corollary 4.1.12.**
A polynomial of degree n with coefficients in the field F
has at most n distinct roots in F.

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