Introduction to Algebraic Number Theory

95 Pages·00.55 MB·English

by  Frédérique Oggier

Introduction to Algebraic Number Theory
F. Oggier
2
A few words
Thesearelecturenotesfortheclassonintroductiontoalgebraicnumbertheory,
given at NTU from January to April 2009 and 2010.
TheselecturesnotesfollowthestructureofthelecturesgivenbyC.Wu¨thrich
at EPFL. I would like to thank Christian for letting me use his notes as basic
material.
I also would like to thank Martianus Frederic Ezerman, Nikolay Gravin and
LIN Fuchun for their comments on these lecture notes.
Attheendofthesenotescanbefoundashortbibliographyofafewclassical
books relevant (but not exhaustive) for the topic: [3, 6] are especially friendly
for a first reading, [1, 2, 5, 7] are good references, while [4] is a reference for
further reading.
3
4
Contents
1 Algebraic Numbers and Algebraic Integers 7
1.1 Rings of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Norms and Traces . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Ideals 19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Factorization and fractional ideals . . . . . . . . . . . . . . . . . 22
2.3 The Chinese Theorem . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Ramification Theory 33
3.1 Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Prime decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Relative Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Normal Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Ideal Class Group and Units 49
4.1 Ideal class group . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Dirichlet Units Theorem . . . . . . . . . . . . . . . . . . . . . . . 53
5 p-adic numbers 57
5.1 p-adic integers and p-adic numbers . . . . . . . . . . . . . . . . . 59
5.2 The p-adic valuation . . . . . . . . . . . . . . . . . . . . . . . . . 62
6 Valuations 67
6.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2 Archimedean places . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.3 Non-archimedean places . . . . . . . . . . . . . . . . . . . . . . . 71
6.4 Weak approximation . . . . . . . . . . . . . . . . . . . . . . . . . 74
5
6 CONTENTS
7 p-adic fields 77
7.1 Hensel’s way of writing. . . . . . . . . . . . . . . . . . . . . . . . 79
7.2 Hensel’s Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.3 Ramification Theory . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.4 Normal extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.5 Finite extensions of Q . . . . . . . . . . . . . . . . . . . . . . . . 88
p
1
Chapter
Algebraic Numbers and Algebraic
Integers
1.1 Rings of integers
We start by introducing two essential notions: number field and algebraic inte-
ger.
Definition 1.1. A number field is a finite field extension K of Q, i.e., a field
which is a Q-vector space of finite dimension. We note this dimension [K : Q]
and call it the degree of K.
Examples 1.1. 1. The field
Q(√2)= x+y√2 x,y Q
{ | ∈ }
is a number field. It is of degree 2 over Q. Number fields of degree 2 over
Qarecalledquadraticfields. Moregenerally,Q[X]/f(X)isanumberfield
if f is irreducible. It is of degree the degree of the polynomial f.
2. Let ζ be a primitive nth root of unity. The field Q(ζ ) is a number field
n n
called cyclotomic field.
3. The fields C and R are not number fields.
Let K be a number field of degree n. If α K, there must be a Q-linear
∈
dependency among 1,α,...,αn , since K is a Q-vector space of dimension n.
{ }
In other words, there exists a polynomial f(X) Q[X] such that f(X) = 0.
∈
We call α an algebraic number.
Definition 1.2. An algebraic integer in a number field K is an element α K
∈
which is a root of a monic polynomial with coefficients in Z.
7
8 CHAPTER 1. ALGEBRAIC NUMBERS AND ALGEBRAIC INTEGERS
Example1.2. SinceX2 2=0,√2 Q(√2)isanalgebraicinteger. Similarly,
− ∈
i Q(i)isanalgebraic integer, sinceX2+1=0. However, anelementa/b Q
∈ ∈
is not an algebraic integer, unless b divides a.
Now that we have the concept of an algebraic integer in a number field, it is
natural to wonder whether one can compute the set of all algebraic integers of
a given number field. Let us start by determining the set of algebraic integers
in Q.
Definition 1.3. The minimal polynomial f of an algebraic number α is the
monic polynomial in Q[X] of smallest degree such that f(α)=0.
Proposition 1.1. The minimal polynomial of α has integer coefficients if and
only if α is an algebraic integer.
Proof. Iftheminimalpolynomialofαhasintegercoefficients,thenbydefinition
(Definition 1.2) α is algebraic.
Now let us assume that α is an algebraic integer. This means by definition
thatthereexistsamonicpolynomialf Z[X]suchthatf(α)=0. Letg Q[X]
∈ ∈
be the minimal polyonial of α. Then g(X) divides f(X), that is, there exists a
monic polynomial h Q[X] such that
∈
g(X)h(X)=f(X).
(Note that h is monic because f and g are). We want to prove that g(X)
actually belongs to Z[X]. Assume by contradiction that this is not true, that
is, there exists at least one prime p which divides one of the denominators of
the coefficients of g. Let u > 0 be the smallest integer such that pug does
not have anymore denominators divisible by p. Since h may or may not have
denominators divisible by p, let v 0 be the smallest integer such that pvh has
≥
no denominator divisible by p. We then have
pug(X)pvh(X)=pu+vf(X).
The left hand side of this equation does not have denominators divisible by p
anymore, thus we can look at this equation modulo p. This gives
pug(X)pvh(X) 0 F [X],
p
≡ ∈
where F denotes the finite field with p elements. This give a contradiction,
p
sincethelefthandsideisaproductoftwonon-zeropolynomials(byminimality
of u and v), and F [X] does not have zero divisor.
p
Corollary 1.2. The set of algebraic integers of Q is Z.
Proof. Let a Q. Its minimal polynomial is X a. By the above proposition,
b ∈ − b
a is an algebraic integer if and only b= 1.
b ±
Definition 1.4. The set of algebraic integers of a number field K is denoted
by . It is usually called the ring of integers of K.
K
O
1.1. RINGS OF INTEGERS 9
The fact that is a ring is not obvious. In general, if one takes a,b two
K
O
algebraic integers, it is not straightforward to find a monic polynomial in Z[X]
which has a+b as a root. We now proceed to prove that is indeed a ring.
K
O
Theorem 1.3. Let K be a number field, and take α K. The two statements
∈
are equivalent:
1. α is an algebraic integer.
2. TheAbeliangroupZ[α]isfinitelygenerated(agroupGisfinitelygenerated
if there exist finitely many elements x ,...,x G such that every x G
1 s
∈ ∈
can be written in the form x = n x +n x +... +n x with integers
1 1 2 2 s s
n ,...,n ).
1 s
Proof. Let α be an algebraic integer, and let m be the degree of its minimal
polynomial, which is monic and with coefficients in Z by Proposition 1.1. Since
all αu with u m can be written as Z-linear combination of 1,α,...,αm 1, we
−
≥
have that
Z[α]=Z Zα ... Zαm 1
−
⊕ ⊕ ⊕
and 1,α,...,αm 1 generate Z[α] as an Abelian group. Note that for this
−
{ }
proof to work, we really need the minimal polynomial to have coefficients in Z,
and to be monic!
Conversely, let us assume that Z[α] is finitely generated, with generators
a ,...,a , where a = f (α) for some f Z[X]. In order to prove that α is
1 m i i i
∈
an algebraic integer, we need to find a monic polynomial f Z[X] such that
∈
f(α)=0. Let N be an integer such that N >degf for i=1,...,m. We have
i
that
m
αN = b a , b Z
j j j
∈
j=1
X
that is
m
αN b f (α)=0.
j j
−
j=1
X
Let us thus choose
m
f(X)=XN b f (X).
j j
−
j=1
X
Clearly f Z[X], it is monic by the choice of N >degf for i=1,...,m, and
i
∈
finally f(α)=0. So α is an algebraic integer.
Example 1.3. We have that
a
Z[1/2]= b is a power of 2
b |
n o
is not finitely generated, since 1 is not an algebraic integer. Its minimal poly-
2
nomial is X 1.
− 2
10 CHAPTER 1. ALGEBRAIC NUMBERS AND ALGEBRAIC INTEGERS
Corollary 1.4. Let K be a number field. Then is a ring.
K
O
Proof. Letα,β . TheabovetheoremtellsusthatZ[α]andZ[β]arefinitely
K
∈O
generated,thussoisZ[α,β]. Now,Z[α,β]isaring,thusinparticularα β and
±
αβ Z[α,β]. SinceZ[α β]andZ[αβ]aresubgroupsofZ[α,β],theyarefinitely
∈ ±
generated. By invoking again the above theorem, α β and αβ .
K
± ∈O
Corollary 1.5. Let K be a number field, with ring of integers . Then
K
O
Q =K.
K
O
Proof. It is clear that if x=bα Q , b Q, α , then x K.
K K
∈ O ∈ ∈O ∈
Now if α K, we show that there exists d Z such that αd (that
K
∈ ∈ ∈ O
is αd = β , or equivalently, α = β/d). Let f(X) Q[X] be the minimal
K
∈ O ∈
polynomialofα. Choosedtobetheleastcommonmultipleofthedenominators
of the coefficients of f(X), then (recall that f is monic!)
X
ddeg(f)f =g(X),
d
(cid:18) (cid:19)
and g(X) Z[X] is monic, with αd as a root. Thus αd .
K
∈ ∈O
1.2 Norms and Traces
Definition 1.5. Let L/K be a finite extension of number fields. Let α L.
∈
We consider the multiplication map by α, denoted by µ , such that
α
µ : L L
α
→
x αx.
7→
ThisisaK-linearmapoftheK-vectorspaceLintoitself(orinotherwords,an
endomorphismoftheK-vectorspaceL). Wecallthenormofαthedeterminant
of µ , that is
α
N (α)=det(µ ) K,
L/K α
∈
and the trace of α the trace of µ , that is
α
Tr (α)=Tr(µ ) K.
L/K α
∈
Note that the norm is multiplicative, since
N (αβ)=det(µ )=det(µ µ )=det(µ )det(µ )=N (α)N (β)
L/K αβ α β α β L/K L/K
◦
while the trace is additive:
Tr (α+β)=Tr(µ )=Tr(µ +µ )=Tr(µ )+Tr(µ )=Tr (α)+Tr (β).
L/K α+β α β α β L/K L/K
In particular, if n denotes the degree of L/K, we have that
N (aα)=anN (α), Tr (aα)=aTr (α), a K.
L/K L/K L/K L/K
∈