Algebraic Number Theory, Study notes of Number Theory

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ALGEBRAIC NUMBER THEORY

Romyar Sharifi

• Introduction
• Part 1. Algebraic number theory
• Chapter 1. Abstract algebra
  • 1.1. Tensor products of fields
  • 1.2. Integral extensions
  • 1.3. Norm and trace
  • 1.4. Discriminants
  • 1.5. Normal bases
• Chapter 2. Dedekind domains
  • 2.1. Fractional ideals
  • 2.2. Dedekind domains
  • 2.3. Discrete valuation rings
  • 2.4. Orders
  • 2.5. Ramification of primes
  • 2.6. Decomposition groups
• Chapter 3. Applications
  • 3.1. Cyclotomic fields
  • 3.2. Quadratic reciprocity
  • 3.3. Fermat’s last theorem
• Chapter 4. Geometry of numbers
  • 4.1. Lattices
  • 4.2. Real and complex embeddings
  • 4.3. Finiteness of the class group
  • 4.4. Dirichlet’s unit theorem
• Chapter 5. Valuations and completions
  • 5.1. Global fields
  • 5.2. Valuations
  • 5.3. Completions 4 CONTENTS
  • 5.4. Extension of valuations
  • 5.5. Local fields
• Chapter 6. Local-global theory
  • 6.1. Semi-local theory
  • 6.2. Ramification revisited
  • 6.3. Differents and discriminants
• Part 2. Cohomology
• Chapter 7. Group cohomology
  • 7.1. Group rings
  • 7.2. Group cohomology via cochains
  • 7.3. Group cohomology via projective resolutions
  • 7.4. Homology of groups
  • 7.5. Induced modules
  • 7.6. Tate cohomology
  • 7.7. Dimension shifting
  • 7.8. Comparing cohomology groups
  • 7.9. Cup products
  • 7.10. Tate cohomology of cyclic groups
  • 7.11. Cohomological triviality
  • 7.12. Tate’s theorem
• Chapter 8. Galois cohomology
  • 8.1. Profinite groups
  • 8.2. Cohomology of profinite groups
  • 8.3. Galois theory of infinite extensions
  • 8.4. Galois cohomology
  • 8.5. Kummer theory
• Part 3. Class field theory
• Chapter 9. Nonarchimedean local fields
  • 9.1. Multiplicative groups
  • 9.2. Tamely ramified extensions
  • 9.3. Ramification groups
• Chapter 10. Class formations - CONTENTS
  • 10.1. Reciprocity maps
  • 10.2. Norm groups
  • 10.3. Class field theory over finite fields
• Chapter 11. Local class field theory
  • 11.1. The Brauer group of a local field
  • 11.2. Local reciprocity
  • 11.3. Norm residue symbols
  • 11.4. The existence theorem
  • 11.5. Class field theory over Qp
  • 11.6. Ramification groups and the unit filtration
• Chapter 12. Global class field theory via ideals
  • 12.1. Ray class groups
  • 12.2. Statements
  • 12.3. Class field theory over Q
  • 12.4. The Hilbert class field
• Chapter 13. Global class field theory via id`eles
  • 13.1. Restricted topological products
  • 13.2. Adeles
  • 13.3. Id`eles
  • 13.4. Statements
  • 13.5. Comparison of the approaches
  • 13.6. Cohomology of the id`eles
  • 13.7. The first inequality
  • 13.8. The second inequality
  • 13.9. The reciprocity law
  • 13.10. Power reciprocity laws

Introduction

At its core, the ancient subject of number theory is concerned with the arithmetic of the integers. The Fundamental Theorem of Arithmetic, which states that every positive integer factors uniquely into a product of prime numbers, was contained in Euclid’s Elements, as was the infinitude of the set of prime numbers. Over the centuries, number theory grew immensely as a subject, and techniques were developed for approaching number-theoretic problems of a various natures. For instance, unique factorization may be viewed as a ring-theoretic property of Z, while Euler used analysis in his own proof that the set of primes is infinite, exhibiting the divergence of the infinite sum of the reciprocals of all primes. Algebraic number theory distinguishes itself within number theory by its use of techniques from abstract algebra to approach problems of a number-theoretic nature. It is also often considered, for this reason, as a subfield of algebra. The overriding concern of algebraic number theory is the study of the finite field extensions of Q, which are known as number fields, and their rings of integers, analogous to Z. The ring of integers O of a number field F is the subring of F consisting of all roots of all monic polynomials in Z[x]. Unlike Z, not all integer rings are UFDs, as one sees for instance by considering the factorization of 6 in the ring Z[

− 5 ]. However, they are what are known as Dedekind domains, which have the particularly nice property that every nonzero ideal factors uniquely as a product of nonzero prime ideals, which are all in fact maximal. In essence, prime ideals play the role in O that prime numbers do in Z. A Dedekind domain is a UFD if and only if it is a PID. The class group of a Dedekind domain is roughly the quotient of its set of nonzero ideals by its nonzero principal ideals, and it thereby serves as something of a measure of how far a Dedekind domain is from being a principal ideal domain. The class group of a number field is finite, and the classical proof of this is in fact a bit of analysis. This should not be viewed as an anomalous encroachment: algebraic number theory draws heavily from the areas it needs to tackle the problems it considers, and analysis and geometry play important roles in the modern theory. Given a prime ideal p in the integer ring O of a number field F, one can define a metric on F that measures the highest power of p dividing the difference of two points in F. If the finite field O/p has characteristic p, then the completion Fp of F with respect to this metric is known as a p-adic field, and the subring Op that is the completion of O is called its valuation ring. In the case that F = Q,

7

8 INTRODUCTION

one obtains the p-adic numbers Qp and p-adic integers Zp. The archimedean fields R and C are also completions of number fields with respect to the more familiar Euclidean metrics, and are in that sense similar to p-adic fields, but the geometry of p-adic fields is entirely different. For instance, a sequence of integers converges to 0 in Zp if and only it is eventually congruent to zero modulo arbitrarily high powers of p. It is often easier to work with p-adic fields, as solutions to polynomial equations can be found in them by successive approximation modulo increasing powers of a prime ideal. The “Hasse principle” asserts that the existence of a solution to polynomial equations in a number field should be equivalent to the existence of a solution in every completion of it. (The Hasse principle does not actually hold in such generality, which partially explains the terminology.) Much of the formalism in the theory of number fields carries over to a class of fields of finite characteristic, known as function fields. The function fields we consider are the finite extensions of the fields of rational functions Fp(t) in a single indeterminate t, for some prime p. Their “rings of integers”, such as Fp[t] in the case of Fp(t), are again Dedekind domains. Since function fields play a central role in algebraic geometry, the ties here with geometry are much closer, and often help to provide intuition in the number field case. For instance, instead of the class group, one usually considers the related Picard group of divisors of degree 0 modulo principal divisors. The completions of function fields are fields of Laurent series over finite fields. We use the term “global field” refer to number fields and function fields in general, while the term “local field” refers to their nonarchimedean completions. An introductory course in algebraic number theory can only hope to touch on a minute but essen- tial fraction of the theory as it is today. Much more of this beautiful edifice can be seen in some of the great accomplishments in the number theory of recent decades. Chief among them, of course, is the proof of Fermat’s last theorem, the statement of which is surely familiar to you. Wiles’ proof of FLT is actually rather round-about. It proceeds first by showing that a certain rational elliptic curve that can be constructed out of a solution to Fermat’s equation is not modular, and then that all (or really, enough) rational elliptic curves are modular. In this latter aspect of the proof are contained advanced methods in the theory of Galois representations, modular forms, abelian varieties, deformation theory, Iwasawa theory, and commutative ring theory, none of which we will be able to discuss.

NOTATION 0.0.1. Throughout these notes, we will use the term ring to refer more specifically to a nonzero ring with unity.

CHAPTER 1

Abstract algebra

In this chapter, we introduce the many of the purely algebraic results that play a major role in algebraic number theory, pausing only briefly to dwell on number-theoretic examples. When we do pause, we will need the definition of the objects of primary interest in these notes, so we make this definition here at the start.

DEFINITION 1.0.1. A number field (or algebraic number field) is a finite field extension of Q.

We have the following names for extensions of Q of various degrees.

DEFINITION 1.0.2. A quadratic (resp., cubic, quartic, quintic, ...) field is a degree 2 (resp., 3, 4, 5, ...) extension of Q.

1.1. Tensor products of fields PROPOSITION 1.1.1. Let K be a field, and let f ∈ K[x] be monic and irreducible. Let M be a field extension of K, and suppose that f factors as (^) ∏mi= 1 f (^) ie iin M[x], where the fi are irreducible and distinct and each ei is positive. Then we have an isomorphism

κ : K[x]/( f ) ⊗K M −→∼

m ∏ i= 1

M[x]/( f (^) ie i)

of M-algebras such that if g ∈ K[x], then κ((g + ( f )) ⊗ 1 ) = (g + ( f (^) ie i))i.

PROOF. Note that we have a canonical isomorphism K[x] ⊗K M −∼→ M[x] that gives rise to the first map in the composition

K[x]/( f ) ⊗K M −→∼ M[x]/( f ) −→∼

m ∏ i= 1

M[x]/( f (^) ie i),

the second isomorphism being the Chinese remainder theorem. The composition is κ. 

We have the following consequence.

LEMMA 1.1.2. Let L/K be a finite separable extension of fields, and let M be an algebraically closed field containing K. Then we have an isomorphism of M-algebras

κ : L ⊗K M −→∼ (^) ∏ σ : L↪→M

M,

11

1.2. INTEGRAL EXTENSIONS 13

for some βi ∈ L and γi ∈ M, with s taken to be minimal. If x 6 = 0, then the γi are L-linearly dependent, so they are K-linearly dependent. In this case, without loss of generality, we may suppose that

γs +

s− 1 ∑ i= 1

αiγi = 0

for some αi in K. Then

x =

s− 1 ∑ i= 0

(βi − αiβs) ⊗ γi,

contradicting minimality. Thus ker ϕ = 0. 

COROLLARY 1.1.7. Let K be a field and L and M be extensions of K both contained in a given algebraic closure of K. Then L and M are linearly disjoint over K if and only if L ⊗K M is a field.

PROOF. Note that LM is a union of subfields of the form K(α, β ) with α ∈ L and β ∈ M. Since α and β are algebraic over K, we have K(α, β ) = K[α, β ], and every element of the latter ring is a K-linear combination of monomials in α and β. Thus ϕ of Proposition 1.1.6 is surjective, and the result follows from the latter proposition. 

COROLLARY 1.1.8. Let K be a field and L and M be finite extensions of K inside a given algebraic closure of K. Then [LM : K] = [L : K][M : K] if and only if L and M are linearly disjoint over K.

PROOF. Again, we have the surjection ϕ : L ⊗K M → LM given by multiplication which is an injection if and only if L and M are linearly disjoint by Proposition 1.1.6. As L ⊗K M has dimension [L : K][M : K] over K, the result follows. 

REMARK 1.1.9. Suppose that L = K(θ ) is a finite extension of K. To say that L is linearly disjoint from a field extension M of K is by Propostion 1.1.1 exactly to say that the minimal polynomial of θ in K[x] remains irreducible in M[x].

We prove the following in somewhat less generality than possible.

LEMMA 1.1.10. Let L be a finite Galois extension of a field K inside an algebraic closure Ω of K, and let M be an extension of K in Ω. Then L and M are linearly disjoint if and only if L ∩ M = K.

PROOF. We write L = K(θ ) for some θ ∈ L, and let f ∈ K[x] be the minimal polynomial of θ. As Gal(LM/M) ∼= Gal(L/(L ∩ M)) by restriction, we have L ∩ M = K if and only if [LM : M] = [L : K]. Since LM = M(θ ), this occurs if and only if f is irreducible in M[x]. The result then follows from Remark 1.1.9. 

1.2. Integral extensions DEFINITION 1.2.1. We say that B/A is an extension of commutative rings if A and B are commu- tative rings such that A is a subring of B.

14 1. ABSTRACT ALGEBRA

DEFINITION 1.2.2. Let B/A be an extension of commutative rings. We say that β ∈ B is integral over A if β is the root of a monic polynomial in A[x].

EXAMPLES 1.2.3. a. Every element a ∈ A is integral over A, in that a is a root of x − a. b. If L/K is a field extension and α ∈ L is algebraic over K, then α is integral over K, being a root of its minimal polynomial, which is monic.

c. If L/K is a field extension and α ∈ L is transcendental over K, then α is not integral over K. d. The element

2 of Q(

2 ) is integral over Z, as it is a root of x^2 − 2. e. The element α = 1 −

√ 5 2 of^ Q(

5 ) is integral over Z, as it is a root of x^2 − x − 1. PROPOSITION 1.2.4. Let B/A be an extension of commutative rings. For β ∈ B, the following conditions are equivalent:

i. the element β is integral over A, ii. there exists n ≥ 0 such that { 1 , β ,... , β n} generates A[β ] as an A-module, iii. the ring A[β ] is a finitely generated A-module, and iv. there exists a finitely generated A-submodule M of B that such that β M ⊆ M and which is faithful over A[β ].

PROOF. Suppose that (i) holds. Then β is a root of a monic polynomial g ∈ A[x]. Given any f ∈ A[x], the division algorithm tells us that f = qg + r with q, r ∈ A[x] and either r = 0 or deg r < deg g. It follows that f (β ) = r(β ), and therefore that f (β ) is in the A-submodule generated by { 1 , β ,... , β deg^ g−^1 }, so (ii) holds. Since this set is independent of f , it generates A[β ] as an A-module, so (iii) holds. Suppose that (iii) holds. Then we may take M = A[β ], which being free over itself has trivial annihilator. Finally, suppose that (iv) holds. Let

M =

n ∑ i= 1

Aγi ⊆ B

be such that β M ⊆ M, and suppose without loss of generality that β 6 = 0. We have

β γi =

n ∑ j= 1

ai jγ (^) j

for some ai j ∈ A with 1 ≤ i ≤ n and 1 ≤ j ≤ n. Consider A-module homomorphism T : Bn^ → Bn represented by (ai j). The characteristic polynomial f (x) ∈ A[x] of T is monic, and f (β ) acts as zero on M. Since M is a faithful A[β ]-module, we must have f (β ) = 0. Thus, β is integral. 

EXAMPLE 1.2.5. The element 12 ∈ Q is not integral over Z, as Z[ 1 , 2 −^1 ,... , 2 −n] for n ≥ 0 is equal to Z[ 2 −n], which does not contain 2−(n+^1 ).

16 1. ABSTRACT ALGEBRA

Finally, if (iii) holds and β ∈ B, then since β B ⊆ B, the element β is integral over a by Proposi- tion 1.2.4. Thus (i) holds. 

We derive the following important consequence. PROPOSITION 1.2.10. Suppose that C/B and B/A are integral extensions of commutative rings. Then C/A is an integral extension as well.

PROOF. Let γ ∈ C, and let f ∈ B[x] be a monic polynomial which has γ as a root. Let B′^ be the subring of B generated over A by the coefficients of f , which is integral over A as B is. By Proposition 1.2.9, the ring B′^ is then finitely generated over A. As B′[γ] is finitely generated over B′ as well, we have B′[γ] is finitely generated over A. Hence, B[γ] is itself an integral extension of A. By definition of an integral extension, the element γ is integral over A. Since γ ∈ C was arbitrary, we conclude that C is integral over A. 

DEFINITION 1.2.11. Let B/A be an extension of commutative rings. The integral closure of A in B is the set of elements of B that are integral over A.

PROPOSITION 1.2.12. Let B/A be an extension of commutative rings. Then the integral closure of A in B is a subring of B.

PROOF. If α and β are elements of B that are integral over A, then A[α, β ] is integral over A by Proposition 1.2.9. Therefore, every element of A[α, β ], including α + β and α · β , is integral over A as well. That is, the integral closure of A in B is closed under addition, additive inverses, and multiplication, and it contains 1, so it is a ring. 

EXAMPLE 1.2.13. The integral closure of Z in Z[x] is Z, since if f ∈ Z[x] is of degree at least 1 and g ∈ Z[x] is nonconstant, then g( f (x)) has degree deg g · deg f in x, hence cannot be 0.

DEFINITION 1.2.14. a. The ring of algebraic integers is the integral closure Z of Z inside C. b. An algebraic integer is an element of Z. DEFINITION 1.2.15. Let B/A be an extension of commutative rings. We say that A is integrally closed in B if A is its own integral closure in B.

DEFINITION 1.2.16. We say that an integral domain A is integrally closed if it is integrally closed in its quotient field.

EXAMPLE 1.2.17. Every field is integrally closed.

PROPOSITION 1.2.18. Let A be an integrally closed domain, let K be the quotient field, and let L be a field extension of K. If β ∈ L is integral over A with minimal polynomial f ∈ K[x], then f ∈ A[x].

1.2. INTEGRAL EXTENSIONS 17 PROOF. Since β ∈ L is integral, it is the root of some monic polynomial g ∈ A[x] such that f divides g in K[x]. As g is monic, every root of g in an algebraic closure K containing K is integral over K. As every root of f is a root of g, the same is true of the roots of f. Write f = (^) ∏ni= 1 (x − βi) for βi ∈ K integral over A. As the integral closure of A in K is a ring, it follows that every coefficient of f is integral over A, being sums of products of the elements βi. Since f ∈ K[x] and A is integrally closed, we then have f ∈ A[x]. 

The following holds in the case of UFDs.

PROPOSITION 1.2.19. Let A be a UFD, let K be the quotient field of A, and let L be a field extension of K. Suppose that β ∈ L is algebraic over K with minimal polynomial f ∈ K[x]. If β is integral over A, then f ∈ A[x].

PROOF. Let β ∈ L be integral over A, let g ∈ A[x] be a monic polynomial of which it is a root, and let f ∈ K[x] be the minimal polynomial of β. Since f divides g in K[x] and A is a UFD with quotient field K, there exists d ∈ K such that d f ∈ A[x] and d f divides g in A[x]. Since f is monic, d must be an element of A (and in fact may be taken to be a least common denominator of the coefficients of f ). The coefficient of the leading term of any multiple of d f will be divisible by d, so this forces d to be a unit, in which case f ∈ A[x]. 

COROLLARY 1.2.20. Every unique factorization domain is integrally closed.

PROOF. The minimal polynomial of an element a of the quotient field K of a UFD A is x − a. If a ∈/ A, it follows from Proposition 1.2.19 that a is not integral over A. 

EXAMPLES 1.2.21. The ring Z is integrally closed.

EXAMPLE 1.2.22. The ring Z[

17 ] is not integrally closed, since α = 1 +

√ 17 2 is a root of the monic polynomial x^2 − x − 4. In particular, Z[

17 ] is not a UFD.

PROPOSITION 1.2.23. Let B/A be an extension of commutative rings, and suppose that B is an integrally closed domain. Then the integral closure of A in B is integrally closed.

PROOF. Let A denote the integral closure of A in B, and let Q denote the quotient field of A. Let α ∈ Q, and suppose that α is integral over A. Then A[α] is integral over A, so A[α] is integral over A, and therefore α is integral over A. That is, α is an element of A, as desired. 

EXAMPLE 1.2.24. The ring Z of algebraic integers is integrally closed.

PROPOSITION 1.2.25. Let A be an integral domain with quotient field K, and let L be an algebraic extension of K. Then the integral closure B of A in L has quotient field equal to L inside L. In fact, every element of L may be written as bd for some d ∈ A and b ∈ B.

1.3. NORM AND TRACE 19

for some α 1 ,... , αd ∈ K. Then the characteristic polynomial of mα is f s, and we have

NL/K (α) =

d ∏ i= 1

αis and TrL/K (α) = s

d ∑ i= 1

αi.

PROOF. We claim that the characteristic polynomial of the K-linear transformation mα is f s. First suppose that L = K(α). Note that { 1 , α,... , αd−^1 } forms a K-basis of K(α), and with respect to this basis, mα is given by the matrix

A =

0 −a 0 1 0 −a 1

...... ... 1 0 −ad− 2 1 −ad− 1

where ai ∈ K for 1 ≤ i ≤ d are such that

f = xd^ +

d− 1 ∑ i= 0

aixi.

Expanding the determinant of xI − A using its first row, we see that

char mα = det(xI − A) = x det(xI − A′) + (− 1 )d−^1 a 0 det

− 1 x

...... − 1 x − 1

= x det(xI − A′) + a 0 ,

where A′^ is the ( 1 , 1 )-minor of A. By induction on the dimension of A, we may assume that

det(xI − A′) = xd−^1 +

d− 2 ∑ i= 0

ai+ 1 xi,

so char mα = f. Since f = xd^ − tr(mα )xd−^1 + · · · + (− 1 )d^ det(mα ),

we have by expanding out the factorization of f in K[x] that NL/K α and TrL/K α are as stated in this case. In general, if {β 1 ,... , βs} is a basis for L/K(α), then {βiα j^ | 1 ≤ i ≤ s, 0 ≤ j ≤ d − 1 } is a basis for L/K. The matrix of mα with respect to this basis (with the lexicographical ordering on the pairs (i, j)) is the block diagonal matrix consisting of s copies of A. In other words, char mα is the f s, from which the result now follows easily. 

We can also express the norm as a power of a product of conjugates and the trace as a multiple of a sum of conjugates.

20 1. ABSTRACT ALGEBRA

PROPOSITION 1.3.4. Let L/K be a finite extension of fields, and let m = [L : K]i be its degree of inseparability. Let S denote the set of embeddings of L fixing K in a given algebraic closure of K. Then, for α ∈ L, we have

NL/K (α) = (^) ∏ σ ∈S

σ αm^ and TrL/K (α) = m (^) ∑ σ ∈S

σ α.

PROOF. The distinct conjugates of α in a fixed algebraic closure K of K are exactly the τα for τ in the set T of distinct embeddings of K(α) in K. These τα are the distinct roots of the minimal polynomial of α over K, each occuring with multiplicity the degree [K(α) : K]i of inseparability of K(α)/K. Now, each of these embeddings extends to [L : K(α)]s distinct embeddings of L into K, and each extension σ ∈ S of τ sends α to τ(α). By Proposition 1.3.3, we have

NL/K α = (^) ∏ τ∈T

(τα)[L:K(α)][K(α):K]i^ = (^) ∏ σ ∈S

σ α[L:K]i^ ,

and similarly for the trace. 

We have the following immediate corollary.

COROLLARY 1.3.5. Let L/K be a finite separable extension of fields. Let S denote the set of embeddings of L fixing K in a given algebraic closure of K. Then, for α ∈ L, we have

NL/K (α) = (^) ∏ σ ∈S

σ α and TrL/K (α) = (^) ∑ σ ∈S

σ α.

We also have the following.

PROPOSITION 1.3.6. Let M/K be a finite field extension and L be an intermediate field in the extension. Then we have

NM/K = NL/K ◦ NM/L and TrM/K = TrL/K ◦ TrM/L.

PROOF. We prove this for norm maps. Let S denote the set of embeddings of L into K that fix K, let T denote the set of embeddings of M into K that fix L, and let U denote the set of embeddings of M into K that fix K. Since [M : K]i = [M : L]i · [L : K]i, it suffices by Proposition 1.3.4 to show that

∏ δ ∈U

δ α = (^) ∏ σ ∈S

σ

∏ τ∈T

τα

We extend each σ to an automorphism ˜σ of K fixing K. We then have

∏ σ ∈S

σ

∏ τ∈T

τα

= (^) ∏ σ ∈S

∏ τ∈T

( σ˜ ◦ τ)α.

For the trace map, we simply replace the products by sums.