Topics in algebra Chapter IV: Commutative rings and modules I - 1

Lemma 2.32 If R is a U.F.D. then for every finite set {a1, . . . , an} of R
there exists a greatest common divisor.
Definition 2.33 If d is a greatest common divisor of a finite set {a1, . . . , an},
then we use d = (a1, . . . , an) for this fact. If (a1, . . . , an) = 1, then we say
that a1, . . . , an are relatively prime.
Definition 2.34 Let D be a U.F.D. and f =
n
aix i ∈ D[x] be a poly-
nomial. A greatest common divisor of the coefficients a0, . . . , an is called a
content of f and is denoted C(f). f is said to be primitive if C(f) is a unit.
In particular, if f is monic then f is primitive.
Lemma 2.35 (Gauss lemma) If D is a U.F.D. and f, g ∈ D[x], then C(fg) =
C(f)C(g). In particular, the product of any two primitive polynomials is
primitive.
Proof. Since f = C(f)f1 and g = C(g)g1 with f1, g1 primitive, C(fg) =
C(C(f)f1C(g)g1) = C(f)C(g)C(f1g1). Therefore, it suffices to show that
n
f1g1 is primitive. Let f1 = aix i m
and g1 = bix i m+n
; then f1g1 = ckx k ,
with ck =
i+j=k
i=0
i=0
i=0
k=0
aibj. If f1g1 is not primitive, then there is a prime element
p ∈ R such that p|ck for all k. Since C(f1) is a unit, there is an integer s
such that
p|ai for i < s and p ∤ as.
Similarly, there is an integer t such that
p|bi for i < t and p ∤ bt.
Since p is a factor of cs+t = b0as+t+b1as+t−1+· · ·+btas+· · ·+bs+t−1a1+bs+ta0,
asbt ∈ (p), a contradiction. Therefore, f1g1 is primitive.
Lemma 2.36 Let D be a U.F.D. with quotient field F and let f and g be
primitive polynomials of D[x] such that f and g are associate in F [x]; then
f and g are associate in D[x].
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