Polynomial rings

2.5.4 Polynomial rings

2.187 Definition If R is a commutative ring, then a polynomial in the indeterminate x over the ring R is an expression of the form

f (x) = a n x n + ···+a 2 x 2 +a 1 x+a 0 where each a i ∈ R and n ≥ 0. The element a i is called the coefficient of x i in f (x).

The largest integer m for which a m 6= 0 is called the degree of f(x), denoted deg f(x);

a m is called the leading coefficient of f (x). If f (x) = a 0 (a constant polynomial) and

a 0 6= 0, then f(x) has degree 0. If all the coefficients of f(x) are 0, then f(x) is called the zero polynomial and its degree, for mathematical convenience, is defined to be −∞. The polynomial f (x) is said to be monic if its leading coefficient is equal to 1.

2.188 Definition If R is a commutative ring, the polynomial ring R[x] is the ring formed by the set of all polynomials in the indeterminate x having coefficients from R. The two opera- tions are the standard polynomial addition and multiplication, with coefficient arithmetic performed in the ring R.

2.189 Example (polynomial ring) Let f (x) = x 3 + x + 1 and g(x) = x 2 + x be elements of

the polynomial ring Z 2 [x]. Working in Z 2 [x],

f (x) + g(x) = x 3 +x 2 +1

and

f (x)

5 +x · g(x) = x 4 +x 3 + x.

For the remainder of this section,

F will denote an arbitrary field. The polynomial ring

F [x] has many properties in common with the integers (more precisely, F [x] and Z are both Euclidean domains, however, this generalization will not be pursued here). These similar- ities are investigated further.

2.190 Definition Let f (x) ∈ F [x] be a polynomial of degree at least 1. Then f(x) is said to be irreducible over

F if it cannot be written as the product of two polynomials in F [x], each of positive degree.

2.191 Definition (division algorithm for polynomials) If g(x), h(x) ∈ F [x], with h(x) 6= 0, then ordinary polynomial long division of g(x) by h(x) yields polynomials q(x) and r(x) ∈

F [x] such that g(x) = q(x)h(x) + r(x), where deg r(x) < deg h(x). Moreover, q(x) and r(x) are unique. The polynomial q(x) is called the quotient, while

r(x) is called the remainder. The remainder of the division is sometimes denoted g(x) mod

h(x), and the quotient is sometimes denoted g(x) div h(x) (cf. Definition 2.82).

2.192 Example (polynomial division) Consider the polynomials g(x) = x 6 +x 5 +x 3 +x 2 +x+1 and h(x) = x 4 +x 3 + 1 in Z 2 [x]. Polynomial long division of g(x) by h(x) yields

g(x) = x 2 h(x) + (x 3 + x + 1). Hence g(x) mod h(x) = x 3 + x + 1 and g(x) div h(x) = x 2 .

c 1997 by CRC Press, Inc. — See accompanying notice at front of chapter.

§ 2.5 Abstract algebra 79

2.193 Definition If g(x), h(x) ∈ F [x] then h(x) divides g(x), written h(x)|g(x), if g(x) mod h(x) = 0.

Let f (x) be a fixed polynomial in F [x]. As with the integers (Definition 2.110), one can define congruences of polynomials in

F [x] based on division by f (x). 2.194 Definition If g(x), h(x) ∈ F [x], then g(x) is said to be congruent to h(x) modulo f(x)

if f (x) divides g(x) − h(x). This is denoted by g(x) ≡ h(x) (mod f(x)). 2.195 Fact (properties of congruences) For all g(x), h(x), g 1 (x), h 1 (x), s(x) ∈ F [x], the fol-

lowing are true. (i) g(x) ≡ h(x) (mod f(x)) if and only if g(x) and h(x) leave the same remainder

upon division by f (x). (ii) (reflexivity) g(x) ≡ g(x) (mod f(x)). (iii) (symmetry) If g(x) ≡ h(x) (mod f(x)), then h(x) ≡ g(x) (mod f(x)). (iv) (transitivity) If g(x) ≡ h(x) (mod f(x)) and h(x) ≡ s(x) (mod f(x)), then

g(x) ≡ s(x) (mod f(x)). (v) If g(x) ≡g 1 (x) (mod f (x)) and h(x) ≡h 1 (x) (mod f (x)), then g(x) + h(x) ≡

g 1 (x) + h 1 (x) (mod f (x)) and g(x)h(x) ≡g 1 (x)h 1 (x) (mod f (x)). Let f (x) be a fixed polynomial in F [x]. The equivalence class of a polynomial g(x) ∈

F [x] is the set of all polynomials in F [x] congruent to g(x) modulo f (x). From properties (ii), (iii), and (iv) above, it can be seen that the relation of congruence modulo f (x) par- titions

F [x] into equivalence classes. If g(x) ∈ F [x], then long division by f(x) yields unique polynomials q(x), r(x) ∈ F [x] such that g(x) = q(x)f(x) + r(x), where deg r(x) < deg f (x). Hence every polynomial g(x) is congruent modulo f (x) to a unique polyno- mial of degree less than deg f (x). The polynomial r(x) will be used as representative of the equivalence class of polynomials containing g(x).

2.196 Definition

F [x]/(f (x)) denotes the set of (equivalence classes of) polynomials in F [x] of degree less than n = deg f (x). Addition and multiplication are performed modulo f (x).

2.197 Fact

F [x]/(f (x)) is a commutative ring. 2.198 Fact If f (x) is irreducible over F , then F [x]/(f (x)) is a field.