Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed. © 1996

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§ 4.1 Fields; Roots of Polynomials

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 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 nonzero element a in F, the equations

a·x = 1     and     x·a = 1

have a solution x in F, called a multiplicative inverse of a, and denoted by a-1.

Definition 4.1.4. Let F be a field. For am, am-1 , . . . , a1, a0 in F, an expression of the form

amxm + am-1xm-1 + · · · + a1x + a0

is called a polynomial over F in the indeterminate x with coefficients am, am-1, . . . , a0. The set of all polynomials with coefficients in F is denoted by F[x].
If n is the largest nonnegative integer such that an 0, then we say that the polynomial

f(x) = anxn + · · · + a0

has degree n, written deg(f(x)) = n, and an is called the leading coefficient of f(x).
    If the leading coefficient is 1, then f(x) is said to be monic.

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) = amxm + · · · + a0 and c is an element of F, then f(c) = amcm + · · · + a0. 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 Z5 and consider the polynomials x5 -2x + 1 and 4x + 1. For any c in Z5, by Fermat's theorem we have c5 c (mod 5), and so

c5 -2c + 1 -c + 1 4c + 1 (mod 5),

which shows that x5 -2x + 1 and 4x + 1 are identical, as functions.

For the polynomials

f(x) = amxm + am-1xm-1 + · · · + a1x + a0


g(x) = bnxn + bn-1xn-1 + · · · + b1x + b0,

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

ambnxn+m + · · · + (a2b0 + a1b1 + a0b2)x2 + (a1b0 + a0b1)x + a0b0.

The coefficient ck of xk in f(x)g(x) can be described by the formula

ck = ai bk-i.

This definition of the product is consistent with what we would expect to obtain using a naive approach: Expand the product using the distributive law repeatedly (this amounts to multiplying each term be every other) and then collect similar terms.

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) | (xk - ck).

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) = q1(x)(x - c) + k, where q1(x) is in F[x] and k is in F, then q1(x) = q(x) and k = f(c).

Definition 4.1.10. Let f(x) = amxm + · · · + a0 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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