5 Polynomials over Rings and Fields
Definition 5.1 Let R be a commutative ring. (As always we as sume R has an
identity .) R[x]
denotes the ring of polynomials in the variable x with coefficients in R.
Exercise 5.2 If R is an integral domain then R[x] is an integral domain.
Definition 5.3 A unique factorization domain (UFD) is an integral
domain in which
every element can be written uniquely as a product of ir reducible elements . The
factorization
is unique up to the order of the factors and multiplying each factor by units.
Example 5.4 Every field is a UFD. The rings Z and Z[i] are UFDs.
Exercise 5.5 Prove: if R is a UFD then R[x] is a UFD.
Exercise 5.6 For an integral domain R, define the field of quotients (or
field of fractions)
by copying how Q arises from Z. (Hint:
“ rational numbers ” are
equivalence classes of fractions).
Definition 5.7 Let F be a field and F(x) be the field of quotients of F[x].
F(x) is called the
function field over F.
Exercise 5.8 d
is countably infinite
and is uncountably infinite. (Hint:
Show 
is linearly independent over R.)
Definition 5.9 Let F be a field. f ∈ F[x] is irreducible if
f is not constant (i. e., deg f ≥ 1)
and 
(f = gh => deg g = 0 or deg h = 0).
Definition 5.10 Let L/K be a field extension; let α ∈ L. We
say that α is algebraic over
K if 
(f ≠ 0 and f( α) = 0). We define
as the g.c.d. of all such polynomials.
and we call it the minimal polynomial
of α (over K). If all elements of L are algebraic
over K then we call L/K an algebraic extension .
Exercise 5.11 
.
Exercise 5.12 
is irreducible over K.
Exercise 5.13 Every finite extension is algebraic.
Definition 5.14 Let R be a ring. I
R is a left ideal
of R if I is an additive subgroup of R
and 
. Right ideals are defined ana logously . I
is an ideal if it is both a left-and
a right-ideal
Definition 5.15 Let R be a ring and I and ideal of
R. The additive quotient group R/I with
elements a + I is a ring under the multiplication rule (a + I)(b + I) = ab + I.
It is called the
quotient ring.
Exercise 5.16 Let F be a field and f ∈ K[x]. The ring K[x]/(f) if a
field if and only if f is
irreducible.
Definition 5.17 A simple extension K(α ) is the smallest field
containing K and α.
Exercise 5.18 If α is algebraic over K then
.
Exercise 5.19 Let 
be a finite field of
order q. Then
Exercise
5.20 Let q = pn be a prime power . Let
be the product of of all monic
irreducible polynomials of degree d over 
.
Prove that 
. (For this
exercise, do not assume the existence of . The
field = Z/pZ of course exists.)
Exercise 5.21
Let 
be the number of monic irreducible
polynomials of degree d over .
Observe from the preceding exercise that 
.
Infer:
Conclude that 
≠ 0.
Exercise 5.22 Prove that there exists a field of order pn.
Hint. The preceding exercise shows
that there exists an irreducible polynomial f of degree n over
. Take the field
.
Exercise 5.23 Prove: the field of order pk is unique (up to
isomorphism).
6 Irreducibility over Z, Gauss lemma, cyclotomic polynomials
Exercise 6.1 (Gauss Lemma) A polynomial f ∈ Z[x] is primitive
if the g.c.d. of its
coefficients is 1. Prove: the product of primitive polynomials is primitive.
(Hint. Assume
fg = ph where f, g, h ∈ Z[x] and p is a prime. Look at this equation
modulo p and use the
fact that 
is an integral domain.)
Exercise 6.2 If f ∈ Z[x] splits into factors
of lower degree over Q[x] then such a split occurs
over Z[x]. In fact, if f = gh where g, h ∈ Q[x] then
(rg ∈ Z[x] and h/r ∈ Z[x]).
Exercise 6.3 Let 
where the
are distinct
integers. Then f(x) is
irreducible over Q. Hint. Let f = gh where g, h ∈ Z[x]. Observe that
.
Exercise 6.4 Let 
where the
are distinct
integers. Then f(x) is
irreducible over Q. Hint. Let f = gh where g, h ∈ Z[x]. Observe that
±1). Observe further that g never changes sign ; nor does h.)
Exercise 6.5 Let 
. If p is a prime and
,
then f is irreducible over Q. (Hint: Unique factorization in
).
Exercise 6.6 If p is a prime then 
is
irreducible over Q.
(Hint: Introduce the variable z = x - 1.)
Exercise 6.7 g.c.d. 
where d = g.c.d.
(k, l).
Definition 6.8 The n-th cyclotomic polynomial is defined as
where the product extends over all complex primitive n-th
roots of unity .
Exercise 6.9
Exercise 6.10 Let f, g ∈ Z[x] with the
leading coefficient of g equal to 1. If 
then
Exercise 6.11
Exercise 6.12
Exercise 6.13 Let f, g be polynomials over the
field F. Prove: if g2 | f then g | f', where f'
is the (formal) derivative of f.
Exercise 6.14 Prove: if 
then xn
− 1 has no multiple factors over
.
Exercise 6.15
Let a ≠ b and n be positive integers . Let p be a prime. Assume p | g.c.d.
.
Prove: p | g.c.d. (a, b).
Exercise 6.16 Prove: if f is a polynomial over any field of
characteristic p, then f(xp) =
(f(x))p.
Exercise 6.17
Let ω be a complex primitive n-th root of unity. Prove: if p is a prime and
then the minimal polynomials of ω and ωp
(over Q) coincide. (Hint. Let f and g be the
minimal polynomials of ω and ωp, respectively. Assume f ≠ g; then fg
| xn − 1. Observe that
f(x) | g(xp). Look at this equation over
and conclude that xn
−1 has a multiple factor over
, a contradiction.)
Exercise 6.18 A major result is now immediate:
is irreducible over Q.
