Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

446 ALGEBRAIC SYSTEMS [APP. B

EXAMPLE B.15

(a) The ringZof integers is the classical example of a unique factorization domain. The units ofZare 1 and−1.

The only associates ofn∈Zarenand−n. The irreducible elements ofZare the prime numbers.

(b) The setD={a+b

√

13 |a,bintegers}is an integral domain. The units ofDfollow:

± 1 , 18 ± 5

√

13 , − 18 ± 5

√

13

The elements 2, 3−

√

13 and− 3 −

√

13 are irreducible inD. Observe that

4 = 2 · 2 =( 3 −

√

13 )(− 3 −

√

13 )

ThusDis not a unique factorization domain. (See Problem B.97.)

B.7Polynomials Over a Field

This section investigates polynomials whose coefficients come from some integral domain or fieldK.
In particular, we show that polynomials over a fieldKhave many of the same properties as the integers.

Basic Definitions

LetKbe an integral domain or a field. Formally, a polynomialfoverKis an infinite sequence of elements
fromKin which all except a finite number of them are 0; that is,

f=(..., 0 ,an,...,a 1 ,a 0 ) or, equivalently, f (t)=antn+···+a 1 t+a 0

where the symboltis used as an indeterminate. The entryakis called thekth coefficient off.Ifnis the largest
integer for whicha=0, then we say that the degree offisn, written deg(f )=n. We also callanthe leading
coefficient off.Ifan=1, we callfamonicpolynomial. On the other hand, if every coefficient offis 0 thenfis
called thezeropolynomial, writtenf≡0. The degree of the zero polynomial is not defined.
LetK[t]be the collection of all polynomialsf(t)overK. Consider the polynomials

f(t)=antn+···+a 1 t+a 0 and g(t)=bmtm+···+b 1 t+b 0

Then the sumf+gis the polynomial obtained by adding corresponding coefficients; that is, ifm≤n, then

f(t)+g(t)=antn+···+(am+bm)tm+···+(a 1 +b 1 )t+(a 0 +b 0 )

Furthermore, the product offandgis the polynomial

f(t)g(t)=(anbm)tn+m+···+(a 1 b 0 +a 0 b 1 )t+(a 0 b 0 )

That is,

f(t)g(t)=cn+mtn+m+···+c 1 t+c 0 where ck=

∑k

i= 0

aibk−i=a 0 bk+a 1 bk− 1 +···+akb 0

The setKof scalars is viewed as a subset ofK[t]. Specifically, we identify the scalara 0 ∈Kwith the
polynomial
f(t)=a 0 or a 0 =(···, 0 , 0 ,a 0 )

Then the operators of addition and scalar multiplication are preserved by this identification. Thus, the mapping
ψ:K→K[t]defined byψ(a 0 )=a 0 is an isomorphism which embedsKintoK[t].