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 7→ 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. (Zm, +) is a finite abelian group. (Zp, ∗) is a finite abelian group, where p is a prime integer. Hamiltonian Group Q8 = {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 r if ∃a ∈ G s.t. G = [{a}] = {a : r ∈ Z}. Example 2.1.3. (Z, +) is an infinite cyclic group generated by 1 of −1. (Zn, +) 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 gk 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−1b ∈ H

• (aH = bH) ⇔ b−1a ∈ 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 = {aH : a ∈ G} /H 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 = {gH : g ∈ G} /H 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.

/ = {p + m : p ∈ } Z mZ Z Z

p + mZ = {p + mx : x ∈ Z} =p ¯ ∈ Zm

/ = {p¯ : p ∈ } Z mZ Z Thus

/ = m. Z mZ Z Deﬁnition 2.1.9 (Simple Group). A group which has no proper normal subgroup is called a simple group. e.g. (Zp, +) 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 (G1,.) and (G2, ∗) be groups, then a map f : G1 7→ G2 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 epimor- phism.

• If f is an bijective group homomorphism, then it is called a isomorphism and we write G1 ≈ G2.

• A group homomorphism f : G1 7→ G1 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 G1 which are mapped to the identity in G2.

ker(f) = {g ∈ G1 : f(g) = e2} and the image of f to be

im(f) = {f(g): g ∈ G1}

The kernel is a normal subgroup of G1 and and the image is a subgroup of G2.

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 (Zm, +). 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 G1 and G2 be two groups and f : G1 7→ G2 be a surjective group homomorphism, then G ≈ G 1/ker(f) 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 (Zm, +) 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 (Z2, +), (Z3, +), (Z5, +), (Z7, +), respectively. There are only two groups of order 4 (up to isomorphism), namely (Z4, +), which is a cyclic group and (V4, o), which is a non cyclic abelian group. There are only two groups of order 6 (up to isomorphism), namely (Z6, +), which is a cyclic group and S3, 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 opera- tions 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 6= 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. a2 = a for all a ∈ R.

Example 2.2.1. (Z, +,.), (Q, +,.), (R, +,.), (C, +,.) are commutative e rings with identity. √ √ Example 2.2.2. (Zm, +,.), (Z [i] , +,.), Z 2 , +,. , Z −5 , +,. , are commutative integral domains with identity.

Proposition 2.2.1. (Zm, +,.) 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 (Zm) = {r¯ ∈ Zm :(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) = pn.

Remark. The order of finite division ring or a finite field is pn, where p = char (R).

Deﬁnition 2.2.4 (Ring homomorphism). Let R1 and R2 be two rings. A map f : R1 7→ R2 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 con- verse is not true.

Proposition 2.2.4. Let f : R1 7→ R2 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 ∈ R1 Note. Under ring homomorphism image of identity(multiplicative) need not be identity. Eg. f : Z 7→ Z × Z as f (0) = (m, 0) .

Proposition 2.2.5. Let R1 be the ring with identity 1 and f : R1 7→ R2 be surjective ring homomorphism, then f (1) will be identity of R2.

9 Proposition 2.2.6. Let f : R1 7→ R2 be a ring homomorphism, and R2 is an integral domain. If 1 is the identity of R1 then f (1) will be identity of R2. Deﬁnition 2.2.5 (Embedding). An injective ring homomorphism f : R1 7→ R2 is called an embedding and R2 is called an extension of R1. 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 ¯ a the form (a, b) = 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 a F = { :(a, b) ∈ R × R∗} b Corollary. The quotient field of a commutative integral domain is the min- imal 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 1 1 1 ∈ Z & 2 ∈ Q but 2 .1 = 2 ∈/ Z.

Theorem 2.2.9. Let f : R1 7→ R2 be a ring homomorphism. Then,

1. If S1 is a subring of R1, then f (S1) will be subring of R2.

−1 2. If S2 is a subring of R2, then f (S2) will be subring of R1.

3. ker (f) is a subring of R1.

Theorem 2.2.10. Let f : R1 7→ R2 be a ring homomorphism. Then,

1. If S1 is a left ideal of R1 and f is a surjection, then f (S1) will be left ideal of R2.

−1 2. If S2 is a left ideal of R2, then f (S2) will be left ideal of R1.

3. ker (f) is a left of R1. Theorem 2.2.11. Let A be the ideal of a ring R, then R = {x + A : x ∈ /A 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 iﬀ K = B where B is a ideal of R containing /A /A A.

• B = B ⇒ B = B . 1/A 2/A 1 2 • B ∩ B = B ∩ B . 1 2/A 1/A 2/A Theorem 2.2.12 (Fundamental theorem of Ring Homomorphism). Let f : R1 7→ R2 be a surjective ring homomorphism, then

R ≈ R 1/ker(f) 2

11 Theorem 2.2.13 (First Isomorphism Theorem). Let A and B be the R /A ideal of a ring R, such that A ⊂ B, then R/ ≈ . B B /A Theorem 2.2.14 (Second Isomorphism Theorem). Let A and B be the A A+B subrings of a ring R and B is a ideal of R, then 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 is a field. /M Theorem 2.2.16. An ideal P of a commutative ring R is a prime ideal of R iff R is an integral domain. /P 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 Zp 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 Zp and Q .

12 Definition 2.2.11 (Polynomial ). Let R be a ring, then an ordered subset of (a1, a2, ...., an, ....) of R is called a polynomial over R if ∃n ∈ N ∪ {0} such that an 6= 0 and ai = 0∀i > n. n is called the degree of the polynomial and an is called leading coefficient of the polynomial. Two polynomials (a1, a2, ...., an, ....) and (b1, b2, ...., bm, ....) are called equal iff m = n ∧ ai = bi∀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 (a1, a2, ...., an, ....) 0 2 n with leading coefficient an is represented by a0x + a1x + a2x + .... + anx , 0 1 2 n where x , x , x , ...., x represents the coordinates of a0, a1, a2, ...., an respectively and called indeterminate, having the properties- • ax + bx = (a + b) x • xrxs = xr+s = xs+r • x0 behaves as a.x0 = a Definition 2.2.12 (Addition and Multiplication of polynomials). Let n 2 n X i p (x) = a0 + a1x + a2x + .... + anx = aix i=0 m 2 m X j q (x) = b0 + b1x + b2x + .... + bmx = bjx j=0 be polynomials over ring R. Define addition and multiplication of polynomials as

max(m+n) X k p (x) + q (x) = (ak + bk) x k=0 (m+n) X k X p (x) .q (x) = ckx , where ck = aibj k=0 i+j=k . Theorem 2.2.17. Let R be a ring then 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.x0.

• The map φ : R 7→ R [x] deﬁned as φ (a) = a.x0∀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−1a. 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 | m,´ b | m´ ⇒ m | m´

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 T 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 com- ∗ mutative integral domain and δ is a map from R to N ∪ {0}, is called a euclidean domain if given a, b 6= 0 ∈ R there exists q, r ∈ R such that

a = bq + r where r = 0 or else δ (r) < δ (b).

Example 2.4.1. (Z, | |) where | | is a absolute value function ,(Z[i], δ) where 2 2 2 2 δ(a + bi) = a + b ,(Z[ω], δ) where δ(a + bw) = a − ab + b ,(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 6= 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

F [x] E = < 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 : F1 7→ F2 be an isomorphism of ﬁelds. Let p1(x) ∈ F1[x] be an irreducible polynomial and let p2(x) ∈ F2[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 F1) and let β be a root of p2(x) (in some extension of F2). Then there is an isomorphism σ : F1(α) 7→ F2(β) mapping α → β and extending φ,such that σ restricted to F1 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]/ < minα,F (x) > so, [F (α): F ] = deg(minα,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 6= 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 6∈ F , minα,F (x) is of degree 2. So K = F (α). 2 Let minα,F (x) = x + bx + c where b, c ∈ F , then √ −b ± b2 − 4c α = 2 p F (α) = F ( b2 − 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 − α1)(x − α2)...... (x − αn), where αi ∈ 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 xn−1). Consider a polynomial xn−1 ∈ Q[x]. Roots of the polynomial are 2πik 2πk 2πk exp( ) = cos( ) + i sin( ) for k = 0, 1, ...... , (n − 1) n n n Let 2πi ζ = exp( ) n n Then all the other roots are power of ζn. 2πik exp( ) = ζ k n n n Then the splitting field of x − 1 over Q is Q(ζn).

19 th Definition 3.2.1. The filed Q(ζn) is called cyclotomic field of n root of unity.

Theorem 3.2.3. Let φ : F1 7→ F2 be an isomorphism of fields. Let f1(x) ∈ F1[x] be a polynomial and let f2(x) ∈ F2[x] be the polynomial obtained by applying φ to the coefficients of f1(x). Let E1 be a splitting field for f1(x) over F1 and let E2 be a splitting field for f2(x) over F2. Then the isomorphism φ extends to an isomorphism σ : E1 7→ E2, i.e., σ restricted to F1 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. 2 Thus if f(x) = (x − 1) (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

n n−1 f(x) = anx + an−1x + ... + a1x + ao ∈ F [x] is deﬁned to be the polynomial

n−1 n−2 Dxf(x) = nanx + (n − 1)an−1x + ... + 2a2x + a1 ∈ F [x]

Proposition 3.3.1. Let g be the greatest common divisor of f and Dxf .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 xp ie f ∈ F [xp]. 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 pth power in K, i.e., K = Kp. 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 pn n pn−1 x − x over Fp. This polynomial has derivative p x − 1 = −1.So this polynomial is separable, hence has precisely pn roots. The set F consisting n pn of p distinct roots of x − x over Fp will be the splitting field of Fp. Further if F is any field of char p, having dimension pn over it prime n ∗ field Fp. Then F has precisely p elements. and since F is a cyclic group, we have n αp −1 = 1 so n αp = α for every α 6= 0 ∈ F But this means α is a root of xP n − x, hence F is contained in a splitting field for this polynomial. Since we have seen that the splitting field has order pn and splitting fields are unique up to isomorphism, this proves that finite fields of any order pn exist and are unique up to isomorphism. We n shall denote the finite field of order p by Fpn .

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 7→ E : σ is an automorphism } forms a group with respect to composition of maps. and

Aut(E/F ) = {σ : E 7→ 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√ (x) = x2 −2. Since ( 2) is a vector √ 2,Q √ Q √ √ 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 7→ F¯ extends to an isomorphism σ : E 7→ E¯, where E¯ is splitting field ϕ(f(x)).

σ : E −→ E¯ 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 7→ E¯, 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 σ1, σ2..., σn are distinct characters from G to E, then the σi’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 = {σ1, σ2, ..., σn} 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 F1 is a fixed field of Aut(E/F ). Then F ⊆ F1 ⊆ E.By above theorem, [E : F1] = Aut(E/F ). Result follows from the fact [E : F ] = [E : F1][F1 : 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 |   subgroups H |  of K E ←→ of G E  containing F |   |       F   F  given by the correspondence E −→ {element of G fixing E} {the fixed field of H} −→ H which are inverse to each other, under this correspondence

1.( inclusion reversing)If E1, E2 correspond to H1, H2, respectively then E1 ⊂ E2, if and only if H2 ≤ H1. 2. [K : E] = |H| and [E : F ] = [G : H], index of H in G:

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 isomor- phisms 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 E1 , E2 correspond to H1, H2, respectively, then the intersection E1 ∩ E2 corresponds to the group hH1,H2i generated by H1 and H2 and the composite ﬁeld E1E2 corresponds to the intersection H1 ∩ H2. 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 Ac containing elements of R which are integral over A.

We say that A is integrally closed in R if A = Ac. 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. Ac is a subring of R containing A and (Ac)c = Ac i.e. if we take integral closure of the integral closure, we will get nothing new.

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 ) 7→ E(F ) given by m(α)(β) = αβ

Clearly, m(α) is a F-linear transformation. Let A(α) = [aij(α)] represents m(α) with respect to some basis.

28 We deﬁne norm,NE/F (α) , trace, TE/F (α) and characteristic polynomial,charE/F (x), of α, relative to extension E/F , as follows

NE/F (α) = det m(α) TE/F (α) = trace m(α) and charE/F (α)(x) = det [xI−A(α)]

r Proposition 4.2.1. charE/F (α)(x) = [minα,F (x)] , where r = [E : F (α)].

Corollary. Let [E : F ] = n and [F (α): F ] = d. Let α1, α2, ...., αd be the roots of minα,F (x), counting multiplicity, in a splitting ﬁeld. Then

d d d Y n X Y n N(α) = α ,T (α) = α , char(α)(x) = (x − α ) d i d i i i=1 i=1 i=1 Proof. Result follows from the above theorem and from the fact that

char(α)(x) = xn − T (α)xn−1 + ... + (−1)nN(α)

Proposition 4.2.2. Let E/F be a separable extension of degree n, let σ1, σ2, ..., σn 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 n Y X NE/F (α) = σi(α),TE/F (α) = σi(α) i=0 i=0 n Y charE/F (α)(x) = (x − σi(α)) i=0 Proposition 4.2.3. Let us consider AKLB setup. Let α ∈ B, then the coeﬃcient of minα,F (x) and charE/F (α)(x) are integral over A, In particular TL/K (α) and NL/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 β element a ∈ A and β ∈ B such that α = 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.

29 4.2.1 Discriminant Deﬁnition 4.2.2. Let [L : K] = n, the discriminant of n-tuple α = (α1, α2, ..., αn) of elements of L is

D(α) = det(TL/k(αiαj))

Note. D(α) ∈ K and if αi ∈ B, then D(α) is integral over A i.e. D(α) ∈ B . If A is integrally closed and αi ∈ B, then D(α) ∈ A.

Proposition 4.2.6. Let σ1, σ2, ..., σn be distinct K-embedding of L into an algebraic closure of L, then 2 D(α) = det(σi(αj))

Proposition 4.2.7. Let α = (α1, α2, ..., αn), then the αi will forms a basis of L over K iﬀ D(α) 6= 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, θ2, ....., θn over K, and θ1, θ2, ..., θn are roots of f in a splitting ﬁeld, with θ1 = θ. Then D coincides Q 2 with the i

L—B | | K—A setup, If A is integrally closed, then B is a submodule of free A-module of rank n. If A is a PID, then B itself is free of rank n over A.

Note. The basis of B(A) will also be the basis of L(K).This basis is known as integral basis of L. Integral basis always exists in case L is a number ﬁeld. Also note that the discriminant is same for all integral basis, because they are related by unimodular matrix. This discriminant is called a ﬁeld discriminant.

30 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 I1I2....In, then P contains Ij 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 n1 n2 nr R, then I can be factored uniquely as P1 P2 ....Pr , a product of prime ideals, where the ni 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 I1 and I2 are integral ideals, we say that I1 divides I2 if I2 = JI1 for some integral ideal J.

Corollary. Let I1 and I2 are integral ideals, then I1 divides I2 iﬀ I1 ⊆ I2. 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 PB. 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

r Y ei PB = Pi i=1

Note that Pi ∩ A = P , for P = P ∩ A ⊆ PA ∩ A ⊆ PB ∩ A ⊆ Pi ∩ A and P is a maximal ideal. ei is called ramification index of Pi over P . We say that P ramifies in B (or in L) if ei > 1 for at least one i. Proposition 4.4.1. Assuming the above setup, one can identify A/P with a subfield of B/Pi and B/Pi as a finite extension of A/P .

Note. The degree fi of the above extension is called relative degree of Pi 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 X eifi = [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 = hai with a 6= 0, N(I) = NL/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 | = pf(P )

where p is unique rational prime in P and f(P ) = [B/P : Z/pZ], the relative degree of P over hpi. 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.

n Deﬁnition 4.6.1 ( Lattices ). Consider a vector space R over R, with a basis e1, e2, ..., en. Then the Z-module

H = Ze1 + Ze1 + ... + Zen

34 n is called a lattice in R . The fundamental domain of H is given by n n X T = α ∈ R : α = aiei, 0 ≤ ai < 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 n centrally symmetric, convex and Lebesgue measurable subset of R and H be a lattice. If

1. µ(S) > 2nv(H), or

2. µ(S) ≥ 2nv(H) and S is compact, then S ∩ H∗ 6= 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 σ1, σ2, ..., σn be the Q-monomorphisms of L into C. Reordering the Q-monomorphisms so that

real embeddings z }| { σ1, σ2, ...., σr1 σr1+1, σr1+2, ...., σr1+r2 , σr1+r2+1, σr1+r2+2, ...., σr1+2r2 | {z } complex embeddings

σr1+r2+j is complex conjugate of σr1+j and n = r1 + 2r2.

r r Deﬁne a map σ : L 7→ R 1 × C 2 by σ(α) = σ1(α), σ2(α), ...., σr1+r2 (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 n, so is σ(I). Therefore σ(I) is a lattice in R . The volume of fundamental domain of the lattice is p v(σ(I)) = 2−r2 |d|N(I)

In particular ,σ(B) is also a lattice and p v(σ(B)) = 2−r2 |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 (α)| ≤ | d|N(I) L/Q π nn

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 (I)| ≤ | d| L/Q π nn

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.