Abstract Algebra and Algebraic Number Theory

Abstract Algebra
and
Algebraic Number Theory
Shashank Singh
January 23, 2011

Contents
1 Introduction 2
2 Basic Algebraic Structures 3
2.1 Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Ring and Integral Domain . . . . . . . . . . . . . . . . . . . . 7
2.3 Arithmetic in Rings . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Domains{ED,PID,UFD} . . . . . . . . . . . . . . . . . . . . . 15
3 Field Extensions 17
3.1 Algebraic Extension . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Splitting Field and Algebraic Closure . . . . . . . . . . . . . . 19
3.3 Separable Extensions . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Normal Extensions . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 Galois Extension . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Algebraic Number Theory 27
4.1 Algebraic Number and Algebraic Integer . . . . . . . . . . . . 27
4.2 Norms, Traces and Discriminants . . . . . . . . . . . . . . . . 28
4.2.1 Discriminant . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Dedekind Domain . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3.1 Unique Factorization of Ideals . . . . . . . . . . . . . . 32
4.4 Factorization of Primes in Extensions . . . . . . . . . . . . . 32
4.5 Norm of an Ideal . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.6 Ideal Class Group . . . . . . . . . . . . . . . . . . . . . . . . 34
1

Chapter 1
Introduction
The word algebra stems out from the name of a famous book ”Al-Jabr wa-al-
Muqabalah” by an Arab Mathematician Alkarismi. Alkarismi lived around
the year 800 A.D. In this book he described the basic algebraic techniques
to simplify algebraic equations.
In the Modern abstract algebra, we study the algebraic structures such
as groups, rings and fields in the axiomatic and structured way.
Modern abstract algebra arises in attempts to solve the polynomial equations.
There were exact methods to solve the polynomial equations of degree
up to 4. These methods reduce the polynomial into a lower degree auxiliary
equation(s), known as resolvent equation(s). Resolvent equations are then
solved using existing methods.
Lagrange tried to solve quintic during 1770. While analyzing the quintic
Lagrange found that the resolvent equation is of degree six. He did not
succeed. Later, Ruffini and Abel proved the unsolvability of the quintic
using the ideas of Lagrange resolvent.
It was Galois, however, who made the fundamental conceptual advances,
and who is considered by many as the founder of group theory. Galois
described group as a collection of permutations closed under multiplication.
In this short note, we will discuss basic concepts of the Group Theory and
Field Theory and using that we will try to cover some aspects of algebraic
number theory.
Though the Galois’s group concept was slightly different than that we
see now. We will discuss it in more structured and simpler way.
For knowing more about the history of Abstract Algebra, please go
through the book [4]. See the books [3], [1] and [2] for more details of
abstract algebra and algebraic number theory.
2

Chapter 2
Basic Algebraic Structures
2.1 Group
Definition 2.1.1 (Binary Operation). Let G be a set. A binary operation
on G is a function o : GXG ↦→ G.
Definition 2.1.2 (Group). Let G be a non empty set together with a
binary operation o. We say that (G, o) is a group if the following properties
are satisfied.
• Associativity. The binary operation o is associative. i.e. (aob) oc =
ao (boc) ∀a, b, c ∈ G.
• Identity. There is an element e ∈ G, called identity element of the
group G, s.t. aoe = a∀a ∈ G.
• Inverse. For each element a in G, there is an element b ∈ G, called
inverse of a in G, s.t. aob = e = boa.
Note. In addition, if the above binary operation is commutative, i.e aob =
boa, then we call that group Abelian group or Commutative Group.
Example 2.1.1. (Z, +), (Q, +), (R, +), (C, +), (Q, ∗), (R, ∗), (C, ∗) are
infinite abelian groups. (Z m , +) is a finite abelian group. (Z p , ∗) is a finite
abelian group, where p is a prime integer.
Hamiltonian Group Q 8 = {1, −1, i, −i, j, −j, k, −k} is a non-abelian
group.
Definition 2.1.3 (Subgroup). If (G, o) is a group and H is a nonempty
subset of G. We say that H is a subgroup of G, if (H, o) is itself a Group.
3

Proposition 2.1.1. Consider a group (Z, +). A non-empty subset H of Z
is a subgroup of (Z, +) iff H = mZ for some m ∈ Z.
Subgroup Generated by a subset. Let S be a subset of a group
(G, o).Then intersection of all those subgroups of G, which contains S is
also a subgroup of G.
This smallest subgroup of G containing the subset S is called a subgroup
generated by S and is denoted by [S].
Example 2.1.2. [φ] = {e}, [G] = G.
Definition 2.1.4 (Cyclic Group). A group (G, o) is called a cyclic group
if ∃a ∈ G s.t. G = [{a}] = {a r : r ∈ Z}.
Example 2.1.3. (Z, +) is an infinite cyclic group generated by 1 of −1.
(Z n , +) is finite cyclic group generated by 1.
Note. Generator of a cyclic group may not be unique.
The order of a group G is simply the number of elements in G.The
order of an element g in a group is the least positive integer k such, that
g k is the identity if there is such a number k, or infinite otherwise.
Definition 2.1.5 (Coset Decomposition). Let H be a subgroup of G. Let
a ∈ G, then aH = {aoh : h ∈ H} is called left coset of H in G determined
by a. Ha = {hoa : h ∈ H} is called right coset of H in G determined by
a.
Properties
• (aH = H) ⇔ a ∈ H.
• (aH = bH) ⇔ a −1 b ∈ H
• (aH = bH) ⇔ b −1 a ∈ H
• If G is a finite group, then, o (H) = o (aH).
• Number of left cosets of H equals number of right coset H.
Definition 2.1.6 (Index of a subgroup). If a group G is finite, then the
number of left cosets of a subgroup H of a G is called the index of H in G,
denoted by [G : H]. The set of left cosets of H in G is denoted by G /H .
G /H = {aH : a ∈ G}
It is called quotient set of G by the subgroup H.
4

Theorem 2.1.2 (Lagrange). The order and index of a subgroup of a finite
group divide the order of a group.In other words, if H is a subgroup of a
finite group G, then,
o (G) = o (H) . [G : H]
Corollary. Every group of a prime order is cycle and hence abelian.
Definition 2.1.7 (Normal Subgroup). A subgroup, N, of a group, G, is
called a normal subgroup (denoted by N ⊳ G) if family of left cosets is same
as family of right cosets; that is, N ⊳ G ⇔ {gN : g ∈ G} = {Ng : g ∈ G}.
Remark. If G is a abelian group, then every subgroup of G is normal in G.
Theorem 2.1.3. H ⊳ G ⇔ ghg −1 ∈ H∀a ∈ G, ∀h ∈ H
Definition 2.1.8 (Quotient Group). Let H ⊳ G, then
G /H = {gH : g ∈ G}
forms a group with respect to the operation ∗, defined as.
xH ∗ yH = (xoy) H for all x, y ∈ G
This group is called the quotient group of G by the normal subgroup H.
Example 2.1.4. Consider (Z, +) and a subgroup mZ.
Thus
Z /mZ = {p + mZ : p ∈ Z}
p + mZ = {p + mx : x ∈ Z} = ¯p ∈ Z m
Z /mZ = {¯p : p ∈ Z}
Z /mZ = Z m .
Definition 2.1.9 (Simple Group). A group which has no proper normal
subgroup is called a simple group. e.g. (Z p , +) is a simple group.
Definition 2.1.10 (Maximal Normal Subgroup). Proper normal subgroup
H of G is called a maximal normal subgroup of G if K be the normal
subgroup of G such that H ⊂ K then either H = K or K = G. i.e. there is
no proper normal subgroup between H and G.
5

Definition 2.1.11 (Group Homomorphism). Let (G 1 , .) and (G 2 , ∗) be
groups, then a map f : G 1 ↦→ G 2 s.t. f (x.y) = f (x)∗f (y), is called a group
homomorphism.
• If f is an injective group homomorphism, then it is called a monomorphism.
• If f is an surjective group homomorphism, then it is called a epimorphism.
• If f is an bijective group homomorphism, then it is called a isomorphism
and we write G 1 ≈ G 2 .
• A group homomorphism f : G 1 ↦→ G 1 s.t. f (x.y) = f (x) .f (y) is
called an indomorphism and if f is an isomorphism, then it is called
as automorphism.
We define the kernel of f to be the set of elements in G 1 which are
mapped to the identity in G 2 .
and the image of f to be
ker(f) = {g ∈ G 1 : f(g) = e 2 }
im(f) = {f(g) : g ∈ G 1 }
The kernel is a normal subgroup of G 1 and and the image is a subgroup of
G 2 .
Theorem 2.1.4. Every infinite cyclic group is isomorphic to (Z, +).
Theorem 2.1.5. Every finite cyclic group of order m is isomorphic to
(Z m , +).
Proposition 2.1.6. Let (G, o) be a group, then
Aut (G) = {f : fis an automorphism on G }
forms a group with respect to composition of maps.
Theorem 2.1.7 (Fundamental Theorem of group homomorphism).
Let G 1 and G 2 be two groups and f : G 1 ↦→ G 2 be a surjective group homomorphism,
then
G 1/ker(f) ≈ G 2
6

Note (Survey of Groups upto order 7). We know that every finite group
of prime order is cyclic and two finite cyclic group of same order are isomorphic.
Further (Z m , +) is a cyclic group.Then we can say that there is only
one group of order 1, which is {e} , only one groups of order 2, 3, 5, 7 are
(Z 2 , +), (Z 3 , +), (Z 5 , +), (Z 7 , +), respectively.
There are only two groups of order 4 (up to isomorphism), namely
(Z 4 , +), which is a cyclic group and (V 4 , o), which is a non cyclic abelian
group.
There are only two groups of order 6 (up to isomorphism), namely
(Z 6 , +), which is a cyclic group and S 3 , which is a non abelian group.
2.2 Ring and Integral Domain
Definition 2.2.1 (Ring). A nonempty set R along with two binary operations
called addition denoted by a + b and multiplication denoted by ab
is said to be a ring if it satisfies the following properties:
• (R, +) is an abelian group.
• Multiplication is associative, i.e.a (bc) = (ab) c for all a, b, c ∈ R.
• Distributive laws hold: a (b + c) = ab + ac and (b + c) a = ba + ca for
all a, b, c ∈ R.
Definition 2.2.2. Let R be a ring.
• If multiplication in R is commutative, it is called a commutative
ring.
• If there is an identity for multiplication(represented by 1), then R is
said to have ring with identity.
• A nonzero element a ∈ R is said to be unit or invertible in R, if
∃b ∈ R s.t ab = ba = 1. Set of units of of R is represented by U (R).
U (R) forms a group with respect to multiplication of R.
• If 1 ≠ 0 in R, and all nonzero elements are invertible, then R is called
a division ring.
• A commutative division ring is called a field.
• An element a of a commutative ring R is called a zerodivisor if there
is a nonzero b ∈ R such that ab = 0. An element a ∈ R that is not
7

a zerodivisor is called a nonzerodivisor. If all nonzero elements of a
commutative ring are nonzero divisors, then R is called an integral
domain.
• A nonempty subset S of a ring R is called a subring of R if S is a
ring with respect to addition and multiplication in R.
• A ring (R, +, .) is called a zero ring if a.b = 0 for all a, b ∈ R. In
particular {0} is a zero ring.
Remark. Every abelian group can be made a ring which is a zero ring.
• A ring is called as a boolean ring, if all of its elements are idempotent,
i.e. a 2 = a for all a ∈ R.
Example 2.2.1. (Z, +, .), (Q, +, .), (R, +, .), (C, +, .) are commutative e
rings with identity.
Example 2.2.2. (Z m , +, .), (Z [i] , +, .), ( Z [√ 2 ] , +, . ) , ( Z [√ −5 ] , +, . ) , are
commutative integral domains with identity.
Proposition 2.2.1. (Z m , +, .) is an integral domain iff m is a prime integer.
Example 2.2.3.
U (Q) = Q ∗
U (R) = R ∗
U (C) = C ∗
U (Z) = {1, −1}
U (Z [i]) = {1, −1, i, −i}
U (Z m ) = {¯r ∈ Z m : (r, m) = 1}
Theorem 2.2.2. Let (R, =, .) be an integral domain, then each non zero
element of R has same additive order.
Proof. Let a, b ∈ R\{0} and let o (a) = m
Then m is the least positive integer such that
ma = 0
⇒ (ma) .b = 0
⇒ (a + a + − − −m times − − + a) .b = 0
⇒ (ab + ab + − − −m times − − + ab) = 0
⇒ a. (b + b + − − −m times − − + b) = 0
8

⇒ a. (mb) = 0
⇒ mb = 0 so o(b)/o(a). similarly we can show that o(a)/o(b). ⇒ o(a) =
o(b)
Definition 2.2.3 (Characteristic of integral domain). Additive order
of a non zero element of an integral domain R is called the characteristic
of the integral domain. If no such a ∈ R exists s.t. na = 0, we define
char(R) = 0.
Theorem 2.2.3. Characteristic of an integral domain is either zero or a
prime number .
Corollary. If p is the characteristic of a finite integral domain, then p/o (R).
Corollary. If R is a finite integral domain with characteristic p, then o (R) =
p n .
Remark. The order of finite division ring or a finite field is p n , where p =
char (R).
Definition 2.2.4 (Ring homomorphism). Let R 1 and R 2 be two rings.
A map f : R 1 ↦→ R 2 is called a ring homomorphism if,
f (x + y) = f (x) + f (y) ∀x, y ∈ R
f (x.y) = f (x) .f (y) ∀x, y ∈ R
Remark. Every ring homomorphism is a group homomorphism, but the converse
is not true.
Proposition 2.2.4. Let f : R 1 ↦→ R 2 is a ring homomorphism.Then
• f (0) = 0
• f (−a) = −f (a)
• f (m.a) = m.f (a) , m ∈ Z
• f (a − b) = f (a) − f (b) a, b ∈ R 1
Note. Under ring homomorphism image of identity(multiplicative) need not
be identity. Eg. f : Z ↦→ Z × Z as f (0) = (m, 0) .
Proposition 2.2.5. Let R 1 be the ring with identity 1 and f : R 1 ↦→ R 2 be
surjective ring homomorphism, then f (1) will be identity of R 2 .
9

Proposition 2.2.6. Let f : R 1 ↦→ R 2 be a ring homomorphism, and R 2 is
an integral domain. If 1 is the identity of R 1 then f (1) will be identity of
R 2 .
Definition 2.2.5 (Embedding). An injective ring homomorphism f :
R 1 ↦→ R 2 is called an embedding and R 2 is called an extension of R 1 .
Theorem 2.2.7. Every ring can be embedded into a ring with identity.
Theorem 2.2.8. Every commutative integral domain R having more than
one element can be embedded into a field F . The element of the field are of
the form (a, ¯ b) = a b
Note. The field F as constructed above is called a field of fraction or
quotient field of the integral domain R.Thus the quotient field
F = { a b : (a, b) ∈ R × R∗ }
Corollary. The quotient field of a commutative integral domain is the minimal
extension of the integral domain into a field in the sense that if ´F
is another field of extension of the integral domain R, then the field ´F is a
extension of quotient field F .
Remark. Every field is a quotient field of itself since every field is a commutative
integral domain.
Definition 2.2.6 (Ideal of a ring). Let A be a non empty subset of a ring
R.
1. A is called a left ideal of R, if
• (A, +) is a subgroup of (R, +).
• x.a ∈ A, ∀x ∈ R & ∀a ∈ A
2. A is called a right ideal of R, if
• (A, +) is a subgroup of (R, +).
• a.x ∈ A, ∀x ∈ R & ∀a ∈ A
3. A is called an ideal of R, if it is both left ideal and right ideal. i.e.
• (A, +) is a subgroup of (R, +).
• a.x ∈ A and x.a ∈ A, ∀x ∈ R & ∀a ∈ A
10

Note. Every ideal left and right of a ring is a subring but subring need
not be an ideal. eg. Z is subring of Q, but Z is not the ideal of Q as
1 ∈ Z & 1 2 ∈ Q but 1 2 .1 = 1 2
/∈ Z.
Theorem 2.2.9. Let f : R 1 ↦→ R 2 be a ring homomorphism. Then,
1. If S 1 is a subring of R 1 , then f (S 1 ) will be subring of R 2 .
2. If S 2 is a subring of R 2 , then f −1 (S 2 ) will be subring of R 1 .
3. ker (f) is a subring of R 1 .
Theorem 2.2.10. Let f : R 1 ↦→ R 2 be a ring homomorphism. Then,
1. If S 1 is a left ideal of R 1 and f is a surjection, then f (S 1 ) will be
left ideal of R 2 .
2. If S 2 is a left ideal of R 2 , then f −1 (S 2 ) will be left ideal of R 1 .
3. ker (f) is a left of R 1 .
Theorem 2.2.11. Let A be the ideal of a ring R, then R /A = {x + A : x ∈
R} forms a ring with respect to addition and multiplication defined as:
(x + A) + (y + A) = (x + y) + A
(x + A) . (y + A) = (x.y) + A
Note. The ring above is called a quotient ring or difference ring of the
ring R by an ideal A.
Example 2.2.4. Let A be the left ideal of a ring R, then
• K is a left ideal of R /A iff K = B /A where B is a ideal of R containing
A.
• B 1/A = B 2/A ⇒ B 1 = B 2 .
• B 1 ∩ B 2/A = B 1/A ∩ B 2/A .
Theorem 2.2.12 (Fundamental theorem of Ring Homomorphism).
Let f : R 1 ↦→ R 2 be a surjective ring homomorphism, then
R 1/ker(f) ≈ R 2
11

Theorem 2.2.13 (First Isomorphism Theorem). Let A and B be the
ideal of a ring R, such that A ⊂ B, then R /B ≈ R / A
B/ A
.
Theorem 2.2.14 (Second Isomorphism Theorem). Let A and B be the
subrings of a ring R and B is a ideal of R, then
A
A∩B ≈ A+B
B .
Definition 2.2.7 (Left ideal generated by an element of a ring). Let
R be a ring & a ∈ R , then
[a] = {na + ra : n ∈ Z & r ∈ R}
is a left ideal of R. It is called left ideal of R generated by a.In particular if
R is a ring with identity 1, then
[a] = {n (1.a) + ra : n ∈ Z & r ∈ R}
[a] = {(n.1) a + ra : n ∈ Z & r ∈ R}
[a] = {(n.1 + r) a : n ∈ Z & r ∈ R}
[a] = {xa : x ∈ R} = Ra
Definition 2.2.8 (Maximal Ideal). A proper ideal I of a ring R is called
a maximal ideal of R, if there is no proper ideal of R containing I.
Definition 2.2.9 (Prime Ideal). A proper ideal P of a ring R is called
a prime ideal of R, if a ∈ P and b ∈ P ⇒ ab ∈ P .
Theorem 2.2.15. M is a maximal ideal of a commutative ring R with
identity iff R /M is a field.
Theorem 2.2.16. An ideal P of a commutative ring R is a prime ideal of
R iff R /P is an integral domain.
Definition 2.2.10 (Prime Field). A field is called a prime field if it has
no proper subfields.
Example 2.2.5. If p is a prime, then Z p is a finite prime field.Rational field
Q is also a prime field.
Remark. Prime subfield of a field is the field generated by the identities .
Note. There are only two prime fields (up to isomorphism ) namely Z p and
Q .
12

Definition 2.2.11 (Polynomial ). Let R be a ring, then an ordered subset
of (a 1 , a 2 , ...., a n , ....) of R is called a polynomial over R if ∃n ∈ N ∪ {0} such
that a n ≠ 0 and a i = 0∀i > n. n is called the degree of the polynomial
and a n is called leading coefficient of the polynomial.
Two polynomials (a 1 , a 2 , ...., a n , ....) and (b 1 , b 2 , ...., b m , ....) are called equal
iff m = n ∧ a i = b i ∀i ∈ N ∪ {0}.
The polynomial (0, 0, ...., 0, ....), in which each coordinate is zero is called
a zero polynomial over R. In practice degree of zero polynomial is
taken to be inf.
Representation of a polynomial: The polynomial (a 1 , a 2 , ...., a n , ....)
with leading coefficient a n is represented by a 0 x 0 + a 1 x + a 2 x 2 + .... + a n x n ,
where x 0 , x 1 , x 2 , ...., x n represents the coordinates of a 0 , a 1 , a 2 , ...., a n respectively
and called indeterminate, having the properties-
• ax + bx = (a + b) x
• x r x s = x r+s = x s+r
• x 0 behaves as a.x 0 = a
Definition 2.2.12 (Addition and Multiplication of polynomials). Let
p (x) = a 0 + a 1 x + a 2 x 2 + .... + a n x n =
q (x) = b 0 + b 1 x + b 2 x 2 + .... + b m x m =
n∑
a i x i
i=0
m∑
b j x j
be polynomials over ring R. Define addition and multiplication of polynomials
as
.
p (x) + q (x) =
p (x) .q (x) =
(m+n)
∑
k=0
max(m+n)
∑
k=0
Theorem 2.2.17. Let R be a ring then
j=0
(a k + b k ) x k
c k x k , where c k = ∑
i+j=k
a i b j
R [x] = {p (x) : p (x) is a polynomial over R}
forms a ring with respect to the addition and multiplication of a polynomials.
13

Note. Let R be a ring.
• If R is commutative then R [x] is also commutative ring .
• If R is a ring with identity 1, then R [x] is also a ring with identity
1 = 1.x 0 .
• The map φ : R ↦→ R [x] defined as φ (a) = a.x 0 ∀a ∈ R is an embedding.
• R [x] is an integral domain iff R is an integral domain.
• If R is a commutative integral domain with identity then R and R [x]
have same unit elements.
• R [x] can’t be a field, even if R is a field.
2.3 Arithmetic in Rings
Let R be a commutative integral domain with identity and R ∗ denote the set
of non zero elements of R. An element a ∈ R ∗ is said to divide an element
b ∈ R ∗ if there is an element c ∈ R ∗ such that b = ac. We use notation a | b
to say that a divided b and a is said to be divisor or factor of b or b is a
multiple of a.
Note. A unit divide every nonzero element of R as a = uu −1 a.
Definition 2.3.1. a, b ∈ R ∗ are said to be associates if a | b and b | a or
equivalently they differ by a unit. We denote them by a ∼ b.
Note. Units and associates of a ∈ R always divide a.
Definition 2.3.2 (Irreducible Element). A non unit element a ∈ R ∗ is
said to be irreducible element of R if it has no proper divisors.
Definition 2.3.3 (Prime Element). A non unit element p ∈ R ∗ is said to
be prime element of R if p | ab ⇔ p | a ∨ p | b.
Definition 2.3.4 (GCD). A element d ∈ R ∗ is said to be greatest common
divisor divisor of a, b ∈ R ∗ if
1. d | a and d | b
2. ´d | a, ´d | b ⇒ ´d | d
Definition 2.3.5 (LCM). A element m ∈ R ∗ is said to be least common
multiple(LCM) of a, b ∈ R ∗ if
14

1. a | m and b | m
2. a | ḿ, b | ḿ ⇒ m | ḿ
Note. GCD and LCM are unique up to associates.
Proposition 2.3.1. Every prime element is irreducible.
Note. An irreducible element is a ring need not be a prime element.
Consider a ring Z[ √ −5] = {a + b √ −5 : a, b ∈ Z}.Units U(Z[ √ −5]) =
{+1, −1}. The element 2 = 2 + 0. √ −5 is irreducible in Z[ √ −5]. And
2 | (1 + √ −5).(1 − √ −5) = 6 but 2 ∤ (1 + √ −5) and 2 ∤ (1 + √ −5)
So 2 is not a prime element in Z[ √ −5].
Note. GCD and LCM of two elements in a ring may or may not exist.
Proposition 2.3.2. Let R be a commutative integral domain with identity.Then
1. a | b ⇔ Rb ⊆ Ra
2. a ∼ b ⇔ Ra = Rb
3. m is a LCM of a, b ⇔ Rm = Ra ⋂ Rb.
4. d is a gcd of a, b ⇔ Rd is a smallest principal ideal containing a and b.
2.4 Domains{ED,PID,UFD}
Definition 2.4.1 (Euclidean Domain). A pair (R, δ), where R is a commutative
integral domain and δ is a map from R ∗ to N ∪ {0}, is called a
euclidean domain if given a, b ≠ 0 ∈ R there exists q, r ∈ R such that
where r = 0 or else δ (r) < δ (b).
a = bq + r
Example 2.4.1. (Z, | |) where | | is a absolute value function ,(Z[i], δ) where
δ(a + bi) = a 2 + b 2 , (Z[ω], δ) where δ(a + bw) = a 2 − ab + b 2 , (F [x], deg)
where F is field, are Euclidean Domains.
Note. Every field F is an euclidean domain with respect to δ defined by
δ(a) = 1∀a ≠ 0.
15

Remark. Arithmetic properties of an euclidean domain does not depends on
a choice of δ.
Proposition 2.4.1. In ED, gcd exists and every irreducible elements are
prime.
Definition 2.4.2 (Principal Ideal Domains). A commutative integral
domain with 1 is said to be principal ideal domain(PID) if every ideal of R
is a principal ideal.
Example 2.4.2. The ring Z is a PID. Every division ring and hence every
field is PID as there are only two ideals {0} and ring itself, which is generated
by identity.
Proposition 2.4.2. In a PID, GCD exists and every irreducible element is
a prime element.
Theorem 2.4.3. The polynomial ring R[x] is a PID iff R is a field.
Note. Z[x] is not a PID for Z is not a field.
Definition 2.4.3 (Unique Factorization Domain). A commutative integral
domain R with identity 1 is said to be an unique factorization domain(UFD)
if Every nonzero non unit can be expressed as a product of
irreducible elements of R. This representation is unique up to ordering and
associates.
Example 2.4.3. Z is a UFD. Every field is UFD, as as there is no non unit
element.
Proposition 2.4.4. In UFD, GCD exists and every irreducible element is
prime element.
Proposition 2.4.5. Every PID is a UFD.
Proposition 2.4.6. Every ED is UFD.
Theorem 2.4.7 (Gauss). If R is UFD the R[x] is also UFD.
Note. UFD need not be PID. e.g. Z[x] is UFD(from Gauss thm) but it is
not a PID.
16

Chapter 3
Field Extensions
Let F be a subfield of E, then E is said to be an extension of F and is
denoted by E/F or F → E. Note that E will then be a vector space over
field F . Dimension of the vector space E (F ) is called degree of the extension
and is denoted by [E : F ]. Extension is said to be finite if the above degree
is finite.
Note. Every field is an extension of its prime subfield.
Definition 3.0.4 ( Root of a polynomial in an extension ). Let E/F
be a field extension and f (x) ∈ F [x] and let α ∈ E, then α is said to be a
root of the polynomial f (x) if f (α) = 0.
Theorem 3.0.8. Let F be a field and p (x) ∈ F [x] is an irreducible polynomial
of degree greater than 1, then we can find a field E, containing an
isomorphic copy of F , having a root of p (x). Moreover
E =
F [x]
< p (x) >
Remark. All the roots of p (x) is algebraically indistinguishable.
Proof.
Theorem 3.0.9. Let F be a field and let p(x) ∈ F [x] be an irreducible
polynomial. Suppose E is an extension of F containing a root α of p(x).
Let F (α) denote the subfield of E generated over F by α. Then
F (α) ∼ = F [x]/ < p(x) >
17

Theorem 3.0.10. Let φ : F 1 ↦→ F 2 be an isomorphism of fields. Let p 1 (x) ∈
F 1 [x] be an irreducible polynomial and let p 2 (x) ∈ F 2 [x] be the irreducible
polynomial obtained by applying the map φ to the coefficients of p(x). Let
α be a root of p(x) (in some extension of F 1 ) and let β be a root of p 2 (x)
(in some extension of F 2 ). Then there is an isomorphism σ : F 1 (α) ↦→
F 2 (β) mapping α → β and extending φ,such that σ restricted to F 1 is the
isomorphism φ.
3.1 Algebraic Extension
Let F/E be a field extension and α ∈ E. α is said to be an algebraic over
F if it is a root of a polynomial f (x) ∈ F [x]. If α is not algebraic over F , it
is said to be transcendental over F .An extension E/F is said to be algebraic
if every element of E is algebraic over F . Let
I = {f (x) ∈ F [x] : f (α) = 0}
Then I will be an ideal of the PID F [x], so I =< m (x) >, for some
m (x) ∈ I. This m (x) ∈ I can be made monic and unique by dividing the
inverse of leading coefficient of generator. This unique, monic, irreducible
polynomial is called minimum polynomial of the element α ∈ E over F .
Definition 3.1.1. Let E be an extension of F . Let α, β, ., ., ∈ E. Then
smallest subfield of E containing both F and the elements α, β, ., ., ., denoted
by F (α, β, ., ., .), is called the field generated by α, β, ., ., . over F .
Note. If a field E is generated by a single element γ (say) over F , then E/F
is said to be a simple extension and γ is said to be a primitive element of
the extension E/F .
Theorem 3.1.1. Let α be algebraic over the field F and let F (α) be the
field generated by α over F . Then
F (α) ∼ = F [x]/ < min α,F (x) >
so,
[F (α) : F ] = deg(min α,F (x)) = deg(α)
Theorem 3.1.2. Every finite extension is algebraic.
Theorem 3.1.3 (Transitivity). If E is algebraic over F and K is algebraic
over E, then K is algebraic over F .
18

Example 3.1.1 (Quadratic Extension). Let F be a field of characteristic
≠ 2. Any extension E of F of degree 2 is called the quadratic extension of
F .
Note. Let α ∈ E \ F . α satisfies an equation of degree at most 2. Since
it can’t satisfy equation of degree 1 as α ∉ F , min α,F (x) is of degree 2. So
K = F (α).
Let min α,F (x) = x 2 + bx + c where b, c ∈ F , then
α = −b ± √ b 2 − 4c
2
F (α) = F ( √ b 2 − 4c)
Note. Quadratic extensions over Q are called quadratic field.
Theorem 3.1.4. Let K be a quadratic field then there is a unique squire
free integer m , such that K = Q( √ m).
3.2 Splitting Field and Algebraic Closure
An extension K of F is said to be a splitting field of a polynomial f(x) if
f(x) factors completely into linear factors in K(x), but not in E(x), where
E is a proper subfield of K. i.e.
f(x) = λ(x − α 1 )(x − α 2 )........(x − α n ), where α i ∈ K, λ ∈ F
Theorem 3.2.1 (Existence of splitting field). If f(x) ∈ F [x], there
exists a field E, which is a spliting field of a f(x).
Proposition 3.2.2. If f ∈ F [x] and deg(f) = n, then f has a splitting field
K over F with [K : F ] ≤ n!.
Example 3.2.1 (Splitting Field of x n −1). Consider a polynomial x n −1 ∈
Q[x]. Roots of the polynomial are
exp( 2πik
n
) = cos(2πk n ) + i sin(2πk ) for k = 0, 1, ........., (n − 1)
n
Let
ζ n = exp( 2πi
n )
Then all the other roots are power of ζ n .
exp( 2πik
n
) = ζ n k
Then the splitting field of x n − 1 over Q is Q(ζ n ).
19

Definition 3.2.1. The filed Q(ζ n ) is called cyclotomic field of n th root
of unity.
Theorem 3.2.3. Let φ : F 1 ↦→ F 2 be an isomorphism of fields. Let f 1 (x) ∈
F 1 [x] be a polynomial and let f 2 (x) ∈ F 2 [x] be the polynomial obtained by
applying φ to the coefficients of f 1 (x). Let E 1 be a splitting field for f 1 (x)
over F 1 and let E 2 be a splitting field for f 2 (x) over F 2 . Then the isomorphism
φ extends to an isomorphism σ : E 1 ↦→ E 2 , i.e., σ restricted to F 1 is
the isomorphism φ.
Corollary (Uniqueness of Splitting Fields). Any two splitting fields for
a polynomial f(x) ∈ F [x] over a field F are isomorphic.
Definition 3.2.2 (Algebraic Closure). The field ¯F is called an algebraic
closure of F if ¯F is algebraic over F and if every polynomial f(x) ∈ F [x]
splits completely over ¯F .
Note. ¯F contains all the roots of all the polynomials in F [x].
Definition 3.2.3 (Algebraically Closed Field). A field K is said to be
algebraically closed if every polynomial with coefficients in K has a root in
K.
Proposition 3.2.4. Let ¯F be an algebraic closure of F . Then ¯F is algebraically
closed.
Remark. Taking algebraic closure of algebraic closure does not give us any
new field. i.e. ¯F = ¯F .
Remark. K = ¯K iff K is algebraically closed.
Proposition 3.2.5. For any field F there exists an algebraically closed field
K containing F .
Proposition 3.2.6 (Uniqueness of Algebraic Closure). Let K be an
algebraically closed field and let F be a subfield of K. Then the collection of
elements ¯F of K that are algebraic over F is an algebraic closure of F . An
algebraic closure of F is unique up to isomorphism.
Theorem 3.2.7 (Fundamental Theorem of Algebra). The field C is
algebraically closed.
Note. C contains algebraic closure of any of its subfields.e.g. ¯Q ⊂ C.
20

3.3 Separable Extensions
In this section we will discuss the multiplicity of a root of a polynomials in
the extension fields.
Definition 3.3.1. An irreducible polynomial f ∈ F [x] is separable if f has
no repeated roots in a splitting field; otherwise f is inseparable. If f is an
arbitrary polynomial, not necessarily irreducible, then we call f separable if
each of its irreducible factors is separable.
Thus if f(x) = (x − 1) 2 (x − 3) over Q, then f is separable, because the
irreducible factors (x − 1) and (x − 3) do not have repeated roots.
Definition 3.3.2. The derivative of the polynomial
f(x) = a n x n + a n−1 x n−1 + ... + a 1 x + a o ∈ F [x]
is defined to be the polynomial
D x f(x) = na n x n−1 + (n − 1)a n−1 x n−2 + ... + 2a 2 x + a 1 ∈ F [x]
Proposition 3.3.1. Let g be the greatest common divisor of f and D x f .f
has a repeated root in a splitting field if and only if the degree of g is at least
1.
Corollary. Over a field of characteristic zero,every polynomial is separable.
Corollary. Over a field F of prime characteristic p, the irreducible polynomial
f is inseparable if and only if f is the zero polynomial. Equivalently,
f is a polynomial in x p ie f ∈ F [x p ].
Theorem 3.3.2. Over a finite field every polynomial is separable.
Definition 3.3.3 (Separable Extension). If E is an extension of F and
α ∈ E, then α is separable over F if α is algebraic over F and min(α, F ) is
a separable polynomial.
If every element of E is separable over F , we say that E is a separable
extension of F .
Note. Every algebraic extension of a field of characteristic zero or a finite
field is separable.
Definition 3.3.4 (Perfect Field). A field K of characteristic p is called
perfect if every element of K is a p th power in K, i.e., K = K p .
Remark. Any field of characteristic 0 is also called perfect.
21

Note. Every irreducible polynomial over a perfect field is separable.
Example 3.3.1 (Existence and Uniqueness of Finite Fields). Let n >
0 be any positive integer and consider the splitting field of the polynomial
x pn − x over F p . This polynomial has derivative p n x pn −1 − 1 = −1.So this
polynomial is separable, hence has precisely p n roots. The set F consisting
of p n distinct roots of x pn − x over F p will be the splitting field of F p .
Further if F is any field of char p, having dimension p n over it prime
field F p . Then F has precisely p n elements. and since F ∗ is a cyclic group,
we have
α pn −1 = 1
so
α pn = α for every α ≠ 0 ∈ F
But this means α is a root of x P n − x, hence F is contained in a splitting
field for this polynomial. Since we have seen that the splitting field has
order p n and splitting fields are unique up to isomorphism, this proves that
finite fields of any order p n exist and are unique up to isomorphism. We
shall denote the finite field of order p n by F p n .
3.4 Normal Extensions
Definition 3.4.1. The algebraic extension E/F is normal if every irreducible
polynomial over F that has at least one root in E splits over E.
Theorem 3.4.1. The finite extension E/F is normal if and only if E is a
splitting field for some polynomial f ∈ F [x].
Note. If E/F is not normal, we can always enlarge E to produce a normal
extension of F . If C is an algebraic closure of E, then C contains all the
roots of every polynomial in F [x], so C/F is normal. Let us try to look for
a smaller normal extension.
Definition 3.4.2 (Normal Closure). Let E be a finite extension of F .
The smallest normal extension of F that contains E is called the normal
closure of E over F .
22

3.5 Galois Extension
Let E be a field and F ⊂ E. Then
Aut(E) = {σ : E ↦→ E : σ is an automorphism }
forms a group with respect to composition of maps. and
Aut(E/F ) = {σ : E ↦→ E : σ is F -automorphism i.e. σ(a) = a∀a ∈ F }
will be a subgroup of the Aut(E).
Note. Prime subfield P of E is generated by {0, 1}. Since any automorphism
σ takes 1 to 1 and 0 to 0, Aut(E) = Aut(E/P ).
Proposition 3.5.1. Let E/F be a field extension. Aut(K) permutes the
roots of irreducible polynomials in F (x) i.e., if α ∈ E is a root of an irreducible
polynomial f(x) in F (x), then σ(α) is also a root of f(x) for all
σ ∈ Aut(E).
Example 3.5.1. Let Q( √ 2)/Q, if τ ∈ Aut(Q( √ 2)) so τ( √ 2) = ± √ 2, as
there are two roots ± √ 2 of the min √ 2,Q (x) = x2 −2. Since Q( √ 2) is a vector
space over Q with basis {1, √ 2}, Aut(Q( √ 2)) = {I, τ}, where τ( √ 2) = − √ 2
and I is identity automorphism. Since Q is a prime subfield of Q( √ 2).
Aut(Q( √ 2)) = Aut(Q( √ 2)/Q) = {I, τ}.
Size of automorphism group in splitting filed Let f(x) ∈ F [x]
and E be splitting field of F . Theorem 3.2.3 shows that any isomorphism
ϕ : F ↦→ ¯F extends to an isomorphism σ : E ↦→ Ē, where Ē is splitting field
ϕ(f(x)).
σ : E −→ Ē
↿
τ : F (α) −→ ¯F (β)
↿
ϕ : F −→ ¯F
Using induction on [E : F ], it can be shown that number of such extensions
is at most [E : F ], with equality if f(x) is separable over F .
In particular case when F = ¯F , ϕ is an identity map and isomorphism
σ : E ↦→ Ē, becomes F -automorphism and we have a theorem:
23
↿
↿

Theorem 3.5.2. Let E be a splitting field of a polynomial f(x) ∈ F [x], then
|Aut(E/F )| ≤ [E : F ]
with equality if f(x) is separable over F .
Note. The above result is true for any finite extension E/F .
Definition 3.5.1 (Galois Extension). E/F is said to be Galois if |Aut(E/F )| =
[E : F ]. In this case Aut(E/F ) is said to be Galois group of E/F and is
denoted by Gal(E/F ).
Note. Splitting field of a separable polynomial f(x) ∈ F [x] is Galois over F .
Definition 3.5.2. If f(x) ∈ F [x] is separable then Galois group of f(x)
over F is the Galois group of splitting field of f(x) over F .
Lemma 3.5.3 (Dedekind). Let G be a group and E a field. A character
from G to E is a homomorphism from G to the multiplicative group E ∗ .
In particular, an automorphism of E defines a character with G = E ∗ , as
does a monomorphism of E into a field L. Dedekind’s lemma states that if
σ 1 , σ 2 ..., σ n are distinct characters from G to E, then the σ i ’s are linearly
independent over E.
Definition 3.5.3. Let E be a field and X ⊂ Aut(E). Let
F ix(X) = {a ∈ E : τ(a) = a∀τ ∈ Aut(E)}
Then F ix(X) is a subfield of E and is called the fixed field of X.
Theorem 3.5.4. Let G = {σ 1 , σ 2 , ..., σ n } be a subgroup of Aut(E).Then
[E : F ix(E)] = n = |G|
Corollary. Let E/F is a finite extension, then |Aut(E/F )| ≤ [E : F ], with
equality iff F is fixed field of Aut(E/F ). I.e. E/F is Galois iff F is fixed
field of Aut(E/F ).
Proof. Let F 1 is a fixed field of Aut(E/F ). Then F ⊆ F 1 ⊆ E.By above
theorem, [E : F 1 ] = Aut(E/F ). Result follows from the fact [E : F ] = [E :
F 1 ][F 1 : F ].
Corollary. Let G is finite subgroup of Aut(K). Let F = F ix(G).Then
E/F is Galois, with Galois group G.
24

Proof. F is fixed by all the element of Aut(E/F ).
[E : F ] = |G| ≤ |Aut(E/F )| ≤ [E : F ]
Theorem 3.5.5. The field extension E/F is Galois iff E is splitting field of
some separable polynomial over F . Furthermore if this is the case, then
E/F is normal as well.
Note ( Characterization of Galois Extension). We now have 4 characterization
of Galois extension E/F .
1. Splitting field of separable polynomial over F .
2. Field, where F is precisely the set of element fixed by Aut(E/F ).
3. Field with [E : F ] = |Aut(E/F )|.
4. Finite, normal and separable extension.
Theorem 3.5.6 (Fundamental theorem of Galois Theory). Let K/F
is a Galois extension and set G = Gat(K/F ), then there is a bijection
⎧
⎫ ⎧
⎫
K
K
⎪⎨
⎪⎩
subfields E |
of K
E
containing F |
F
given by the correspondence
⎪⎬ ⎪⎨
←→
⎪⎭
{the fixed field of H} −→
⎪⎩
subgroups H |
of G
E
|
F
E −→ {element of G fixing E}
which are inverse to each other, under this correspondence
1. (inclusion reversing)If E 1 , E 2 correspond to H 1 , H 2 , respectively
then E 1 ⊂ E 2 , if and only if H 2 ≤ H 1 .
2. [K : E] = |H| and [E : F ] = [G : H], index of H in G:
H
K
| } |H|
E
| } [G : H]
F
⎪⎬
⎪⎭
25

3. K/E is always Galois with Galois group Gal(K/E) = H
K
| H
E
4. E is Galois over F if and only if H is a normal subgroup in G. If this
is the case, then the Galois group is isomorphic to the quotient group
Gal(E/F ) ∼ = G/H
More generally, even if H is not necessarily normal in G, the isomorphisms
of E (into a fixed algebraic closure of F containing K) which
fix F are in one to one correspondence with the cosets {σH} of H in
G.
5. If E 1 , E 2 correspond to H 1 , H 2 , respectively, then the intersection
E 1 ∩ E 2 corresponds to the group 〈H 1 , H 2 〉 generated by H 1 and H 2
and the composite field E 1 E 2 corresponds to the intersection H 1 ∩ H 2 .
Hence the lattice of subfields of K containing F and the lattice of
subgroups of G are “dual” (the lattice diagram for one is the lattice
diagram for the other turned upside down).
26

Chapter 4
Algebraic Number Theory
In this chapter we will discuss the arithmetics of algebraic number fields,
ring of integers in the number field, the ideals in the ring of integers and
unique factorization of ideal etc. We also study the concept of localization
to complete the number field relative to the metric attached to a prime ideal
of a number field. Finally we conclude the chapter with the description of
Ideal Class Theory.
4.1 Algebraic Number and Algebraic Integer
Let E/F be a field extension we know that α ∈ E is algebraic iff α is root
of a non constant polynomial in F [x].
If α ∈ C is algebraic over Q. Then α is called algebraic number and
any algebraic extension over Q is called a number field.
Let A be a subring of R. β ∈ R is called integral over A if β is root
of a monic polynomial f(x) ∈ A[x].
If β ∈ C is integral over Z, then β is called an algebraic integer.
Theorem 4.1.1. Let A is a subring of R, and let β ∈ R. The following are
equivalent:
1. β is integral over A.
2. The A-module A[x] is finitely generated.
27

3. The element β belongs to a subring B of R such that A ⊆ B and B is
finitely generated A-module.
Definition 4.1.1 (Integral Closure). Let A be subring of R, integral closure
of A in R is the set A c containing elements of R which are integral over
A.
We say that A is integrally closed in R if A = A c . If we say that A
is integrally closed without reference to R, it means A is integrally close in
the field of fraction of R.
Note. A c is a subring of R containing A and (A c ) c = A c i.e.
integral closure of the integral closure, we will get nothing new.
if we take
Proposition 4.1.2. If A is UFD, then A is integrally closed.
Note. Z is integrally closed.
Theorem 4.1.3. If L is an algebraic number field then there exists an algebraic
number θ such that L = Q(θ).
Definition 4.1.2 (Basic Setup for ANT). Let A be an integral domain
with quotient field K, and let L be a finite separable extension of K. Let B
be the set of elements of L that are integral over A, that is, B is the integral
closure of A in L. The diagram below summarizes all the information.
L — B
| |
K — A
As a example, A = Z, K = Q, L is a number field, and B is the ring of
algebraic integers of L. Henceforth, we will refer this as the AKLB setup.
4.2 Norms, Traces and Discriminants
Definition 4.2.1. Let E/F be a field extension of degree n, i.e. E(F ) is a
vector space of dimension n. For each α ∈ E, define a map
m(α) : E(F ) ↦→ E(F ) given by m(α)(β) = αβ
Clearly, m(α) is a F-linear transformation. Let A(α) = [a ij (α)] represents
m(α) with respect to some basis.
28

We define norm,N E/F (α) , trace, T E/F (α) and characteristic polynomial,char E/F (x),
of α, relative to extension E/F , as follows
N E/F (α) = det m(α)
T E/F (α) = trace m(α) and char E/F (α)(x) = det [xI−A(α)]
Proposition 4.2.1. char E/F (α)(x) = [min α,F (x)] r , where r = [E : F (α)].
Corollary. Let [E : F ] = n and [F (α) : F ] = d. Let α 1 , α 2 , ...., α d be the
roots of min α,F (x), counting multiplicity, in a splitting field. Then
N(α) = ( d ∏
i=1
α i
)
, T (α) =
(n
d
d∑ )
α i ,
i=1
{ ∏
d
char(α)(x) = (x − α i ) } n d
i=1
Proof. Result follows from the above theorem and from the fact that
.
char(α)(x) = x n − T (α)x n−1 + ... + (−1) n N(α)
Proposition 4.2.2. Let E/F be a separable extension of degree n, let
σ 1 , σ 2 , ..., σ n be the distinct F-embedding of E into an algebraic closure of
E, or equally well into a normal extension L of F containing E. Then
N E/F (α) =
n∏
σ i (α), T E/F (α) =
i=0
char E/F (α)(x) =
n∑
σ i (α)
i=0
n∏
(x − σ i (α))
Proposition 4.2.3. Let us consider AKLB setup. Let α ∈ B, then the
coefficient of min α,F (x) and char E/F (α)(x) are integral over A, In particular
T L/K (α) and N L/K (α) are integral over A. If A is integrally closed then
coefficient belongs to A.
Corollary. An algebraic integer a ∈ Q must in fact belong to Z.
Proposition 4.2.4. In AKLB setup, let α ∈ L, then there is a non zero
element a ∈ A and β ∈ B such that α = β a
, i.e. L is a fraction field of B.
Proposition 4.2.5. In AKLB setup, there is a basis of L/K consisting
entirely the elements of B.
i=0
29

4.2.1 Discriminant
Definition 4.2.2. Let [L : K] = n, the discriminant of n-tuple α =
(α 1 , α 2 , ..., α n ) of elements of L is
D(α) = det(T L/k (α i α j ))
Note. D(α) ∈ K and if α i ∈ B, then D(α) is integral over A i.e. D(α) ∈ B
. If A is integrally closed and α i ∈ B, then D(α) ∈ A.
Proposition 4.2.6. Let σ 1 , σ 2 , ..., σ n be distinct K-embedding of L into an
algebraic closure of L, then
D(α) = [ det(σ i (α j )) ] 2
Proposition 4.2.7. Let α = (α 1 , α 2 , ..., α n ), then the α i will forms a basis
of L over K iff D(α) ≠ 0.
Proposition 4.2.8. Let L = K(θ), and f be a minimum polynomial of θ
over K. Let D be the discriminant of the basis 1, θ, θ 2 , ....., θ n over K, and
θ 1 , θ 2 , ..., θ n are roots of f in a splitting field, with θ 1 = θ. Then D coincides
with the ∏ i

4.3 Dedekind Domain
Definition 4.3.1. An integral domain satisfying following conditions
1. A is Noetherian ring.
2. A is integrally closed.
3. Every non zero prime ideal of A is maximal ideal.
is called a Dedekind domain
Note. Every PID satisfies the above properties and is therefore a Dedekind
domain.
Theorem 4.3.1. In
L — B
| |
K — A
setup, if A is a Dedekind domain, so is B. In particular, ring of algebraic
integer of number field is a Dedekind domain.
Definition 4.3.2 (Fractional Ideal). Let R be an integral domain with
fraction field K, let I be a R-submodule of K. I is said to be a fraction
ideal of R if rI ⊆ R for some r ∈ R ∗ . r is called denominator of factional
ideal I.
Note. An ordinary ideal of R is fractional ideal with denominator 1.
Definition 4.3.3 (Product of Ideals). Product of two ideals I and J is
the ideal generated by the product set IJ. Similarly we can define a product
of finitely many ideals.
Note. If a prime ideal P contains a product of finitely many ideals I 1 I 2 ....I n ,
then P contains I j for some j.
Proposition 4.3.2. Let R be an integral domain with fraction field K.
1. If I is finitely generated R-submodule of K, then I is a fractional ideal.
2. If R is Noetherian and I is fractional ideal of R, then I is finitely
generated R-submodule of K.
31

3. If I and J are fractional ideal with denominators r and s respectively,
then I ∩ J , I + J and IJ are fractional ideals with respective denominators
r (or s), rs and rs.
Note. Let I be a fractional ideal of R. As I is R-submodule of K = frac(R).
RI ⊆ I = 1I ⊆ RI i.e. RI = I.
Proposition 4.3.3. Let I be a non zero prime ideal of a Dedekind domain
R, Let J = {α ∈ K : αI ⊆ R}, then J is fractional ideal of R , R J and
IJ = R.
4.3.1 Unique Factorization of Ideals
Theorem 4.3.4. If I is a nonzero fractional ideal of the Dedekind domain
R, then I can be factored uniquely as P n 1
1 P n 2 nr
2 ....Pr
, a product of prime
ideals, where the n i are integers.
Note. The set I(R) of non zero fractional ideal of Dedekind domain R forms
a group with respect to the multiplication( product ) of ideals. R act as
identity. J defined above will be inverse of ideal I.
Corollary. A non zero fractional ideal I of a Dedekind domain R is an
integral ideal iff all exponent in the prime factorization of I are non-negative.
Definition 4.3.4. Let I 1 and I 2 are integral ideals, we say that I 1 divides
I 2 if I 2 = JI 1 for some integral ideal J.
Corollary. Let I 1 and I 2 are integral ideals, then I 1 divides I 2 iff I 1 ⊆ I 2 .
Note. In case of ideals DIVIDES MEANS CONTAINS
Theorem 4.3.5. Let I be non zero ideal of a Dedekind domain R and let
a ∈ I ∗ , then I can be generated by two elements , one of which is a.
4.4 Factorization of Primes in Extensions
Consider the AKLB setup
L — B
| |
K — A
32

where A is Dedekind domain with fraction field K. Let P is prime ideal of
A. The lifting(extension) of A to B is the ideal P B. If Q is a prime ideal
of B, then contraction of Q to A is the ideal Q ∩ A.
Using unique factorization theorem we can write
P B =
Note that P i ∩ A = P , for P = P ∩ A ⊆ P A ∩ A ⊆ P B ∩ A ⊆ P i ∩ A and
P is a maximal ideal.
r∏
i=1
P e i
i
e i is called ramification index of P i over P .
in B (or in L) if e i > 1 for at least one i.
We say that P ramifies
Proposition 4.4.1. Assuming the above setup, one can identify A/P with
a subfield of B/P i and B/P i as a finite extension of A/P .
Note. The degree f i of the above extension is called relative degree of P i
over P .
Note. B/P B can be shown to be finitely generated A/P -algebra.
Proposition 4.4.2 ( Ram-Rel Identity ). Assuming the above setup. We
have
r∑
e i f i = [B/P B : A/P ] = n
i=1
4.5 Norm of an Ideal
Definition 4.5.1. Assume the AKLB setup
L — B
| |
K — A
with A = Z. Thus A is Dedekind domain, so is B. Let I be a non zero ideal
of B. Define the norm of I by
N(I) = |B/I|
Proposition 4.5.1. Assuming the above setup
33

1. N(I) if finite.
2. Norm is multiplicative ie N(IJ) = N(I)N(J).
3. If I = 〈a〉 with a ≠ 0, N(I) = N L/Q (a).
4. If N(I) is prime, then I is a prime ideal.
5. N(I) ∈ I, so I contains a unique rational prime(which is a prime
factor of N(I).)
6. If P is a prime ideal of B. Then
N(P ) = |B/P | = p f(P )
where p is unique rational prime in P and f(P ) = [B/P : Z/pZ], the
relative degree of P over 〈p〉.
Proposition 4.5.2. A rational number m can belong to only a finitely many
ideals of B.
Corollary. Only finitely many ideals can have the given norm.
4.6 Ideal Class Group
Assume the AKLB setup
L — B
| |
K — A
with A = Z. We know A i.e. Z is Dedekind domain, so B is also a Dedekind
domain.
Let I(L) be the group of factional ideals of Dedekind domain( ring of algebraic
integers) B and P (L) be the group of factional ideals Bω, ω ∈ L.
P (L) is a normal subgroup of I(B). The quotient group C(L) = I(L)/P (L)
is called ideal class group of L. In this section we use Minkowaski theory
to show that ideal class group is finite in this setup.
Definition 4.6.1 ( Lattices ). Consider a vector space R n over R, with a
basis e 1 , e 2 , ..., e n . Then the Z-module
H = Ze 1 + Ze 1 + ... + Ze n
34

is called a lattice in R n . The fundamental domain of H is given by
T = { α ∈ R n : α =
n∑
a i e i , 0 ≤ a i < 1 }
i=1
If µ be the Lebesgue measure, then the volume µ(T ) of fundamental
domain T will be denoted by v(H) and is called the determinant of the
lattice.
Note. The v(H) does not depend on the particular choice of a basis of the
lattice.
Theorem 4.6.1 ( Minkowski’s Convex Body Theorem ). Let S be
centrally symmetric, convex and Lebesgue measurable subset of R n and H
be a lattice. If
1. µ(S) > 2 n v(H), or
2. µ(S) ≥ 2 n v(H) and S is compact,
then S ∩ H ∗ ≠ φ.
Definition 4.6.2. Consider
L — B
| |
Q — Z
Where L be the number field of degree n over Q and B is ring of algebraic
integer in L. Let σ 1 , σ 2 , ..., σ n be the Q-monomorphisms of L into C.
Reordering the Q-monomorphisms so that
real embeddings
{ }} {
σ 1 , σ 2 , ...., σ r1 σ r1 +1, σ r1 +2, ...., σ r1 +r 2
, σ r1 +r 2 +1, σ r1 +r 2 +2, ...., σ r1 +2r 2
} {{ }
complex embeddings
σ r1 +r 2 +j is complex conjugate of σ r1 +j and n = r 1 + 2r 2 .
Define a map σ : L ↦→ R r 1
× C r 2
by
σ(α) = ( σ 1 (α), σ 2 (α), ...., σ r1 +r 2
(α) )
σ is the injective ring homomorphism, known as canonical embedding.
35

Let I be the non-zero integral ideal of B, then I is a free Z-module of rank
n, so is σ(I). Therefore σ(I) is a lattice in R n . The volume of fundamental
domain of the lattice is
v(σ(I)) = 2 −r 2√
|d|N(I)
In particular ,σ(B) is also a lattice and
v(σ(B)) = 2 −r 2√
|d|
Proposition 4.6.2 (Minkowski Bound on Element Norm). If I is a
nonzero integral ideal of R, then ∃α ∈ I ∗ , such that
( ) 4
r2
( ) n!
|N L/Q (α)| ≤
π n n | √ d|N(I)
Proposition 4.6.3 (Minkowski Bound on Ideal Norm). For every ideal
class of B, there is an ideal I, such that
( ) 4
r2
( ) n!
|N L/Q (I)| ≤
π n n | √ d|
Theorem 4.6.4. The ideal class group of a number field is finite.
Proof. We know that there are only finitely many integral ideals of given
norm and by above proposition we can associate each ideal class with an
ideal whose norm is bounded by a fixed constants. If number of ideal classes
were infinite, we would eventually get some integral ideal in two different
classes, which is a contradiction. Hence ideal class group of number filed S
is finite.
36

Bibliography
[1] Robert B. Ash. A Course In Algebraic Number Theory.
[2] Robert B. Ash. Abstract Algebra: The Basic Graduate Year. 2000.
[3] Davis S. Dummit and Richard M. Foote. Abstract Algebra.
[4] Israel Kleiner. A History of Abstract Algebra.
37