Number Theory: An Introduction to Mathematics

3 Polynomials 101

Consequently we can replacedby a proper divisord′, again not a unit, for which

m′+n′>m+n. Since there exists a divisordfor whichm+nis a maximum, this

yields a contradiction.

Now letf,gbe polynomials inR[t] such thatgdividesfinK[t]. Thusf=gH,

whereH∈K[t]. We can writeH=ab−^1 h 0 ,wherea,bare coprime elements ofR

andh 0 is a primitive polynomial inR[t]. Also

f=c(f)f 0 , g=c(g)g 0 ,

wheref 0 ,g 0 are primitive polynomials inR[t]. Hence

bc(f)f 0 =ac(g)g 0 h 0.

Sinceg 0 h 0 is primitive, it follows that

bc(f)=ac(g).

IfH ∈ R[t], thenb=1andsoc(g)|c(f). On the other hand, ifc(g)|c(f),then

bc(f)/c(g)=a.Since(a,b)=1, this implies thatb=1andH∈R[t]. 

Corollary 18If R is a GCD domain, then R[t]is also a GCD domain. If, moreover,

R is a factorial domain, then R[t]is also a factorial domain.

proofLetKdenote the field of fractions ofR.SinceK[t] is a GCD domain and
R[t]⊆K[t],R[t] is certainly an integral domain. Iff,g∈R[t], then there exists
a primitive polynomialh 0 ∈R[t] which is a greatest common divisor offandgin
K[t]. It follows from Proposition 17 that

h=(c(f),c(g))h 0

is a greatest common divisor offandginR[t].

This proves the first statement of thecorollary. It remains to show that ifRalso

satisfies the chain condition (#), thenR[t] does likewise. But if fn ∈ R[t]and

fn+ 1 |fnfor everyn,thenfnmust be of constant degree for all largen. The second

statement of the corollary now also follows from Proposition 17 and the chain

condition inR. 

It follows by induction that in the statement of Corollary 18 we may replace

R[t] by the ringR[t 1 ,...,tm] of all polynomials in finitely many indeterminates

t 1 ,...,tm with coefficients fromR. In particular, ifKis a field, then any polyno-

mialf ∈K[t 1 ,...,tm] such thatf∈/Kcan be represented as a product of finitely

many irreducible polynomials and the representation is essentially unique.

It is now easy to give examples of GCD domains which are not B ́ezout domains.

LetRbe a GCD domain which is not a field (e.g.,R =Z). Then somea 0 ∈ Ris

neither zero nor a unit. By Corollary 18,R[t] is a GCD domain and, by Proposition 17,

the greatest common divisor inR[t] of the polynomialsa 0 andtis 1. If there existed

g,h∈R[t] such that

a 0 g+th= 1 ,