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ALGEBRAIC NUMBER THEORY Contents Introduction ALGEBRAIC NUMBER THEORY J.S. MILNE Abstract. These are the notes for a course taught at the University of Michigan in F92 as Math 676. They are available at www.math.lsa.umich.edu/∼jmilne/. Please send comments and corrections to me at [email protected]. v2.01 (August 14, 1996.) First version on the web. v2.10 (August 31, 1998.) Fixed many minor errors; added exercises and index. Contents Introduction..................................................................1 The ring of integers 1; Factorization 2; Units 4; Applications 5; A brief history of numbers 6; References. 7. 1. Preliminaries from Commutative Algebra ............................10 Basic definitions 10; Noetherian rings 10; Local rings 12; Rings of fractions 12; The Chinese remainder theorem 14; Review of tensor products 15; Extension of scalars 17; Tensor products of algebras 17; Tensor products of fields 17. 2. Rings of Integers ........................................................19 Symmetric polynomials 19; Integral elements 20; Review of bases of A- modules 25; Review of norms and traces 25; Review of bilinear forms 26; Discriminants 26; Rings of integers are finitely generated 28; Finding the ring of integers 30; Algorithms for finding the ring of integers 33. 3. Dedekind Domains; Factorization .....................................37 Discrete valuation rings 37; Dedekind domains 38; Unique factorization 39; The ideal class group 43; Discrete valuations 46; Integral closures of Dedekind domains 47; Modules over Dedekind domains (sketch). 48; Fac- torization in extensions 49; The primes that ramify 50; Finding factoriza- tions 53; Examples of factorizations 54; Eisenstein extensions 56. 4. The Finiteness of the Class Number ..................................58 Norms of ideals 58; Statement of the main theorem and its consequences 59; Lattices 62; Some calculus 67; Finiteness of the class number 69; Binary quadratic forms 71; 5. The Unit Theorem ......................................................73 Statement of the theorem 73; Proof that UK is finitely generated 74; Com- putation of the rank 75; S-units 77; Finding fundamental units in real c 1996, 1998, J.S. Milne. You may make one copy of these notes for your own personal use. i 0J.S.MILNE quadratic fields 77; Units in cubic fields with negative discriminant 78; Finding µ(K) 80; Finding a system of fundamental units 80; Regulators 80; 6. Cyclotomic Extensions; Fermat’s Last Theorem......................82 The basic results 82; Class numbers of cyclotomic fields 87; Units in cyclo- tomic fields 87; Fermat’s last theorem 88; 7. Valuations; Local Fields ................................................91 Valuations 91; Nonarchimedean valuations 91; Equivalent valuations 93; Properties of discrete valuations 95; Complete list of valuations for Q 95; The primes of a number field 97; Notations 97; Completions 98; Com- pletions in the nonarchimedean case 99; Newton’s lemma 102; Extensions of nonarchimedean valuations 105; Newton’s polygon 107; Locally compact fields 108; Unramified extensions of a local field 109; Totally ramified exten- sions of K 111; Ramification groups 112; Krasner’s lemma and applications 113; A Brief Introduction to PARI 115. 8. Global Fields ...........................................................116 Extending valuations 116; The product formula 118; Decomposition groups 119; The Frobenius element 121; Examples 122; Application: the quadratic reciprocity law 123; Computing Galois groups (the hard way) 123; Comput- ing Galois groups (the easy way) 124; Cubic polynomials 126; Chebotarev density theorem 126; Applications of the Chebotarev density theorem 128; Topics not covered 130; More algorithms 130; The Hasse principle for qua- dratic forms 130; Algebraic function fields 130. Exercises..................................................................132. It is standard to use Gothic (fraktur) letters for ideals: abcmnpqABCMNPQ abcmnpqABCMNP Q I use the following notations: X ≈ YXand Y are isomorphic; ∼ X and Y are canonically isomorphic X = Y or there is a given or unique isomorphism; X =df YXis defined to be Y ,orequalsY by definition; X ⊂ YXis a subset of Y (not necessarily proper). Introduction 1 Introduction An algebraic number field is a finite extension of Q;analgebraic number is an element of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in the number field, the ideals in the ring of integers, the units, the extent to which the ring of integers fails to be have unique factorization, and so on. One important tool for this is “localization”, in which we complete the number field relative to a metric attached to a prime ideal of the number field. The completed field is called a local field — its arithmetic is much simpler than that of the number field, and sometimes we can answer questions by first solving them locally, that is, in the local fields. An abelian extension of a field is a Galois extension of the field with abelian Galois group. Global class field theory classifies the abelian extensions of a number field K in terms of the arithmetic of K; local class field theory does the same for local fields. This course is concerned with algebraic number theory. Its sequel is on class field theory (see my notes CFT). I now give a quick sketch of what the course will cover. The fundamental theorem of arithmetic says that integers can be uniquely factored into products of prime powers: an m =0in Z can be written in the form, r1 ··· rn ± m = up1 pn ,u= 1,pi prime number, ri > 0, and this factorization is essentially unique. Consider more generally an integral domain A.Anelementa ∈ A is said to be a unit if it has an inverse in A;IwriteA× for the multiplicative group of units in A. An element p of A is said to prime if it is neither zero nor a unit, and if p|ab ⇒ p|a or p|b. If A is a principal ideal domain, then every nonzero nonunit element a of A can be written in the form, r1 ··· rn a = p1 pn ,pi prime element, ri > 0, and the factorization is unique up to order and replacing each pi with an associate, i.e., with its product with a unit. Our first task will be to discover to what extent unique factorization holds, or fails to hold, in number fields. Three problems present themselves. First, factorization in a field only makes sense with respect to a subring, and so we must define the “ring of integers” OK in our number field K. Secondly, since unique factorization will in general fail, we shall need to find a way of measuring by how much it fails. Finally, since factorization is only considered up to units, in order to fully understand the arithmetic of K, we need to understand the structure of the group of units UK in OK. Resolving these three problems will occupy the first five sections of the course. The ring of integers. Let K be an algebraic number field. Because K is of finite degree over Q, every element α of K is a root of a monic polynomial n n−1 f(X)=X + a1X + ···+ a0,ai ∈ Q. 2 Introduction If α is a root of a monic polynomial with integer coefficients, then α is called an algebraic integer of K. We shall see that the algebraic integers form a subring OK of K. The criterion as stated is difficult to apply. We shall see that to prove that α is an algebraic integer, it suffices to check that its minimum polynomial (relative to Q) has integer coefficients. Consider for example the field K√ = Q[ d], where d is a square-free integer. The minimum polynomial of α = a + b d, b =0, a, b ∈ Q,is √ √ (X − (a + b d))(X − (a − b d)) = X2 − 2aX +(a2 − b2d). Thus α is an algebraic integer if and only if 2a ∈ Z,a2 − b2d ∈ Z. From this it follows easily that √ √ OK = Z[ d]={m + n d | m, n ∈ Z} if d ≡ 2, 3mod4, and √ 1+ d OK = {m + n | m, n ∈ Z} if d ≡ 1mod4, 2 √ i.e., OK is the set of sums m + n d with m and n either both integers or both half-integers. th Let ζd be a primitive d root of 1, for example, ζd =exp(2πi/d), and let K = Q[ζd]. Then we shall see that O Z { i | ∈ Z} K = [ζd]= miζd mi . as one would hope. Factorization. An element p of an integral domain A is said to be irreducible if it is neither zero nor a unit, and can’t be written as a product of two nonunits. For example, a prime element is (obviously) irreducible. A ring A is a unique factorization domain if every nonzero nonunit element of A can be expressed as a product of irreducible elements in essentially one way. Is OK a unique factorization domain? No, not in general! Infact,weshallseethateachelementofOK can be written as a product of irreducible elements (this is true for all Noetherian rings) — it is the uniqueness that fails. √ For example, in Z[ −5] we have √ √ 6=2· 3=(1+ −5)(1 − −5). √ √ Toseethat2,3,1+ −5, 1 − −5 are irreducible, and no two are associates, we use the norm map √ √ Nm : Q[ −5] → Q,a+ b −5 → a2 +5b2. For α ∈OK,wehave Nm(α)=1 ⇐⇒ αα¯ =1 ⇐⇒ α is a unit. (*) Introduction 3 √ √ If 1 + −5=αβ,thenNm(αβ)=Nm(1+ −5) = 6. Thus Nm(α)=1, 2, 3, or 6. In the first case, α is a unit, the second and third cases don’t occur,√ and in the fourth case β is a unit. A similar argument shows that 2, 3, and 1 − −5 are irreducible. Next note that√ (*) implies that√ associates have the same norm, and so it remains to show that 1 + −5and1− −5 are not associates, but √ √ √ 1+ −5=(a + b −5)(1 − −5) has no solution with a, b ∈ Z.
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