A First Course in Number Theory
Theorem 5.3 The ring C[t] of polynomials with coefficients in the field C of complex numbers is a principal ring
5.2 Derivations
Aderivation on a ringRis a mapD:R→Rsuch that
D(x+y) =D(x) +D(y) (5.1)
and
D(xy) =D(x)y+xD(y) (5.2)
for all x, y ∈ R. Condition (5.1) says that D is a homomorphism of the additive group structure of R. Condition (5.2) implies (Exercise 1) that D(1) = 0 and that, ifx∈Ris invertible, then
D(x−1) =−D(x) x2 .
176 5. TheabcConjecture
Moreover, it follows by induction (Exercise 2) that D(x1· · ·xn) =
n i=1
x1· · ·xi−1D(xi)xi+1· · ·xn
for allx1, . . . , xn∈R.
The next result shows that the derivative is a derivation on a polynomial ring.
Theorem 5.5 LetR be a ring and R[t]the ring of polynomials with coef- ficients inR. DefineD:R[t]→R[t] by
D m
i=0
aiti
= m
i=1
iaiti−1.
ThenD is a derivation on R[t].
Proof. Letf =f(t) =m
i=0aiti andg=g(t) =n
j=0bjtj. It is imme- diate that D(f +g) =D(f) +D(g), and so D is a homomorphism of the additive group of polynomials. Since
f(t)g(t) = m i=0
n j=0
aitibjtj=
m+n
k=0
i+j=k
aibjtk,
we have
D(f g) =
m+n
k=1
k
i+j=k
aibjtk−1
=
m+n
k=1
i+j=k
(i+j)aibjti+j−1
=
m+n
k=1
i+j=k
iaiti−1bjtj+
m+n
k=1
i+j=k
aitijbjtj−1
= m
i=1
n j=0
iaiti−1bjtj+ m i=0
n j=1
aitijbjtj−1
= D(f)g+f D(g).
Therefore,D is a derivation onR[t]. 2
Anintegral domain is a ringR such that if b1, b2 ∈R with b1 = 0 and b2 = 0, thenb1b2 = 0. Corresponding to every integral domain is a field, called the quotient field of R. It consists of all fractions of the form a/b,
where a, b ∈R and b = 0, anda1/b1 =a2/b2 if and only if a1b2 = a2b1. Addition and multiplication of fractions are defined in the usual way: If a1, a2, b1, b2∈Rwithb1= 0 andb2= 0, then b1b2= 0 and
a1 b1
+a2 b2
= a1b2+a2b1 b1b2
and a1 b1 ·a2
b2
= a1a2 b1b2
.
The quotient field ofZisQ. IfF[t] is the ring of polynomials with coeffi- cients in a fieldF, then the quotient field ofF[t] is the fieldF(t) of rational functions with coefficients in F. A careful construction of quotient fields can be found in the Exercises.
Theorem 5.6 Let R be an integral domain with quotient fieldF, and let D be a derivation on R. There exists a unique derivation DF on F such that DF(x) =D(x)for all x∈R.
Proof. Suppose that there exists a derivationDFonFsuch thatDF(a) = D(a) for alla∈R. Let x∈F, x= 0. There exista, b∈R withb= 0 and x=a/b. Since a=bx∈R, it follows that
D(a) =DF(a) =DF(bx) =DF(b)x+bDF(x) =D(b)x+bDF(x), and so
DF a
b
=DF(x) = D(a)−D(b)x
b =D(a)b−aD(b)
b2 . (5.3)
Thus, the derivationDF on F is uniquely determined by the derivationD onR. In Exercise 3 we prove that (5.3) defines a derivation on the quotient field RF.2
LetD be a derivation on the fieldF. Forx∈F×, we define thelogarith- mic derivativeL(x) by
L(x) = D(x) x . Ifx, y∈F×, then
L(xy) = D(xy)
xy =D(x)y+xD(y)
xy = D(x)
x +D(y)
y =L(x) +L(y) and
L x
y
= D(x)
x +D(y−1)
y−1 =D(x)
x −D(y)
y =L(x)−L(y) by Exercise 1.
We now consider polynomials with complex coefficients. A field F is calledalgebraically closedif every nonconstant polynomial with coefficients
178 5. TheabcConjecture
in F has at least one zero in F. By the fundamental theorem of algebra, the fieldCis algebraically closed. Letf(t)∈C[t], and letN0(f) denote the number of distinct zeros off(t). Iff(t) has degreenwith leading coefficient an, thenf(t) factors uniquely in the form
f(t) =an N0(f)
i=1
(t−αi)ni,
where α1, . . . , αN0(f) are the distinct zeros of f, the positive integerni is the multiplicity of the zero αi, and n1+· · ·+nN0(f) = n. If D is the derivation onC[t] defined in Theorem 5.5, then, by Exercise 2,
D(f) =an N0(f)
i=1
ni(t−αi)ni−1
N0(f)
j=1 j=i
(t−αj)nj
and
L(f) = D(f)
f =
N0(f) i=1
ni t−αi. Letg(t) =bm
N0(g)
j=1 (t−βj)mj be a nonzero polynomial inC[t], and con- sider the rational functionf /g∈C(t). Then
L f
g
=L(f)−L(g) =
N0(f) i=1
ni
t−αi −
N0(g) j=1
mj
t−βj. (5.4) This algebraic identity will be used in the next section to prove Mason’s theorem.
Exercises
1. LetD be a derivation on a ringR. Prove thatD(1) = 0 and that, if x∈R is invertible, then
D(x−1) =−D(x) x2 . 2. LetD be a derivation on the ringR. Prove that
D(x1· · ·xn) = n i=1
x1· · ·xi−1D(xi)xi+1· · ·xn
for allx1, . . . , xn∈R.
3. LetRbe an integral domain with quotient fieldF. LetDbe a deriva- tion onR, and define the functionDF onF by (5.3). We shall prove thatDF is a derivation on the quotient fieldF.
(a) Prove that DF is well defined, that is, if a1/b1 = a2/b2, then DF(a1/b1) =DF(a2/b2).
(b) Prove that DF
a1 b1
+a2 b2
=DF a1
b1
+DF
a2 b2
.
(c) Prove that DF
a1
b1 ·a2
b2
=DF a1
b1 a2
b2 +a1
b1DF a2
b2
.
4. LetR be a commutative ring with identity. Amultiplicatively closed subset of R is a subset S such that 1 ∈ S and if s1, s2 ∈ S, then s1s2∈S. We consider the set of ordered pairs of the form (r, s) with r∈Rands∈S. Define a relation on this set as follows:
(r, s)∼(r, s) if s(sr−sr) = 0 for somes∈S.
Prove that this is an equivalence relation.
5. Let S−1Rbe the set of equivalence classes of the relation defined in Exercise 4. We denote the equivalence class of (r, s) by the fraction r/s. We also denote the equivalence class (r,1) byr. Define multipli- cation of fractions as follows:
r1
s1· r2
s2 = r1r2
s1s2.
(a) Prove that this multiplication is well defined, that is, if (r1, s1)∼ (r1, s1) and (r2, s2)∼(r2, s2), then (r1r2, s1s2)∼(r1r2, s1s2).
(b) Prove that multiplication inS−1R is associative and commuta- tive, and that the equivalence class of (1,1) is a multiplicative identity.
(c) Prove that the equivalence class of (s,1) is invertible in S−1R for every s∈S.
(d) Prove that
a s = sa
ss for alla∈R ands, s ∈S.
6. Define addition of fractions inS−1R as follows:
r1 s1
+r2 s2
= s2r1+s1r2 s1s2
.
180 5. TheabcConjecture
(a) Prove that this addition is well defined, that is, if (r1, s1) ∼ (r1, s1) and (r2, s2)∼(r2, s2), then (s2r1+s1r2, s1s2)∼(s2r1+ s1r2, s1s2).
(b) Prove that addition in S−1R is associative and commutative, and that multiplication distributes over addition. Prove that the equivalence class of (0,1) is an additive identity.
7. (Localization) In Exercises 4–6 we proved thatS−1R is a ring. This ring is called thering of fractions ofRby S. We also say thatS−1R is constructed bylocalizingRatS.
(a) Prove that if 0∈S, then S−1R={0}.
(b) Prove that if R is an integral domain and 0∈S, thenS−1Ris an integral domain.
(c) Prove that if R is an integral domain and S is the set of all nonzero elements ofR, thenS−1Ris a field. This field is called thequotient field of the integral domainR.
8. Define ϕS:R→S−1R byϕS(r) =r/1 =r.
(a) Prove thatϕS is a ring homomorphism.
(b) Prove that if R is an integral domain and 0 ∈ S, then ϕR is one-to-one.
(c) Prove that ifRis an integral domain andS =R×, thenS−1R is isomorphic toR.
Hint:IfS is a multiplicative subset ofR ands∈S∩R×, then (r, s)∼(s−1r,1) for allr∈R.
9. Let S ={1,2,4,8, . . .} be the multiplicative subset of Zconsisting of the powers of 2. Describe the ring of fractionsS−1Z. What is the group of units in this ring?
10. Let S={±1,±3,±5,±7, . . .}be the multiplicative subset of Zcon- sisting of the odd integers.
(a) Describe the ring of fractionsS−1Z.
(b) Describe the principal ideal generated by 2 in this ring.
(c) Prove that every element of the ring not in this ideal is a unit in S−1Z, and so2is a maximal ideal inS−1Z.
11. Letpbe a prime number and letSbe the set of all integers not divis- ible byp. Prove thatS is a multiplicative subset of Z, and describe the ring of fractionsS−1Z. Prove that the principal ideal generated bypis a maximal ideal inS−1Z.
12. Let F[t] be the polynomial ring with coefficients in the field F. Let S = {1, t, t2, t3, . . .} be the multiplicative subset of F[t] consisting of the powers oft. Prove that S−1F[t] is isomorphic to the ring of Laurent polynomialswith coefficients inF,that is, the ring consisting of all expressions of the formn
i=maiti, whereai∈F, andmandn are integers withm≤n, and addition and multiplication are defined in the usual way.
13. We consider the ring R = Z/12Z, and denote the congruence class a+ 12Zbya
(a) Prove thatS={1,3,9} is a multiplicative subset ofR.
(b) Let ϕS : R → S−1R be the ring homomorphism constructed in Exercise 8. Prove that ϕS(a) = ϕS(b) if and only if a ≡ b (mod 4).
(c) Prove that 1/3 = 3 inS−1R.
(d) Prove thatS−1R∼=Z/4Z.
14. Letm≥2. We consider the ringR=Z/mZ, and denote the congru- ence classa+mZby a. LetS be a multiplicative subset of R such that 0∈S.
(a) Prove that we can factor m uniquely in the form m = m0m1, where (m0, m1) = 1, and ifpis a prime number that dividesm, thenpdividesm0if and only if there is a congruence classs∈S such thatpdividess. Show that (s, m1) = 1 for alls∈S.
(b) Prove that there is a congruence class s0 ∈ S such that m0 divides s0.
(c) Let ϕS : R → S−1R be the ring homomorphism constructed in Exercise 8. Prove that ϕS(a) = ϕS(b) if and only if a ≡ b (mod m1).
(d) Prove that for everys∈S there existsr∈Rsuch that 1/s=r in S−1R.
Hint: Ifs ∈S, then there exists an integer r such thatrs ≡1 (mod m1).
(e) Prove thatS−1R∼=Z/m1Z.