Algebraic number theory Nicolas Mascot ([email protected]), Aurel Page ([email protected]) TAs: Chris Birkbeck ([email protected]), George Turcas ([email protected]) Version: March 20, 2017 Contents 1 Number fields 6 1.1 Resultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Field extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.2 Algebraic elements, algebraic extensions . . . . . . . . 8 1.2.3 The degree of an extension . . . . . . . . . . . . . . . . 12 1.2.4 The trace, norm, and characteristic polynomial . . . . 14 1.2.5 Primitive elements . . . . . . . . . . . . . . . . . . . . 16 1.3 Complex embeddings . . . . . . . . . . . . . . . . . . . . . . . 18 1.3.1 Extension of complex embeddings . . . . . . . . . . . . 18 1.3.2 The signature of a number field . . . . . . . . . . . . . 19 1.3.3 Traces and norms vs. complex embeddings . . . . . . . 19 2 Algebraic integers 22 2.1 The ring of integers . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.1 Monic polynomials . . . . . . . . . . . . . . . . . . . . 22 2.1.2 The ring of integers . . . . . . . . . . . . . . . . . . . . 23 2.2 Orders and discriminants . . . . . . . . . . . . . . . . . . . . . 24 2.2.1 Linear algebra over Z . . . . . . . . . . . . . . . . . . . 24 2.2.2 Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.3 Discriminants, part I . . . . . . . . . . . . . . . . . . . 28 2.3 Computing the maximal order . . . . . . . . . . . . . . . . . . 32 2.3.1 Denominators vs. the index . . . . . . . . . . . . . . . 32 2.3.2 Discriminants, part II . . . . . . . . . . . . . . . . . . . 33 2.4 The case of quadratic fields . . . . . . . . . . . . . . . . . . . 37 2.5 The case of cyclotomic fields . . . . . . . . . . . . . . . . . . . 38 1 3 Ideals and factorisation 40 3.1 Reminder on finite fields . . . . . . . . . . . . . . . . . . . . . 41 3.2 Reminder on ideals . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3 Integral closure . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4 Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . . 45 3.5 Factorisation theory in Dedekind domains . . . . . . . . . . . 47 3.6 Decomposition of primes . . . . . . . . . . . . . . . . . . . . . 50 3.7 Practical factorisation . . . . . . . . . . . . . . . . . . . . . . 52 3.8 Ramification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.9 The case of quadratic fields . . . . . . . . . . . . . . . . . . . 60 3.10 The case of cyclotomic fields . . . . . . . . . . . . . . . . . . . 62 4 The class group 65 4.1 UFDs. vs. PID. vs. Dedekind domains . . . . . . . . . . . . . 65 4.2 Ideal inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3 The class group . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.4 Finiteness of the class group: the Minkowski bound . . . . . . 69 4.5 Applications: Diophantine equations . . . . . . . . . . . . . . 71 4.5.1 Sums of two squares . . . . . . . . . . . . . . . . . . . 71 4.5.2 Another norm equation . . . . . . . . . . . . . . . . . . 72 4.5.3 A norm equation with a nontrivial class group . . . . . 73 4.5.4 Mordell equations . . . . . . . . . . . . . . . . . . . . . 75 4.5.5 The regular case of Fermat’s last theorem . . . . . . . 77 5 Units 78 5.1 Units in a domain . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2 Units in Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 K 5.3 Roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.3.1 Roots of unity under complex embeddings . . . . . . . 81 5.3.2 Bounding the size of W . . . . . . . . . . . . . . . . . 83 K 5.4 Dirichlet’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 84 5.5 The case of quadratic fields . . . . . . . . . . . . . . . . . . . 86 5.6 The case of cyclotomic fields . . . . . . . . . . . . . . . . . . . 88 5.7 The Pell–Fermat equation . . . . . . . . . . . . . . . . . . . . 89 5.8 Class groups of real quadratic fields . . . . . . . . . . . . . . . 90 2 6 Geometry of numbers 93 6.1 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.2 Minkowski’s theorem . . . . . . . . . . . . . . . . . . . . . . . 97 6.3 Applications to number theory . . . . . . . . . . . . . . . . . . 98 7 Summary of methods and examples 103 7.1 Discriminant and ring of integers . . . . . . . . . . . . . . . . 103 7.2 Factorisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.3 Class group and units . . . . . . . . . . . . . . . . . . . . . . . 104 7.4 Complete examples . . . . . . . . . . . . . . . . . . . . . . . . 107 3 Introduction: why algebraic number theory? Consider the following statement, today known as Fermat’s last theorem: Theorem 0.0.1. Let n (cid:62) 3 be an integer. Then the Diophantine equation xn +yn = zn has no nontrivial solution, i.e. xn +yn = zn, x,y,z ∈ Z =⇒ xyz = 0. In 1847, while this theorem still had not been proved, the French math- ematician Gabriel Lam´e had an idea for the case where n is an odd prime. Suppose we have xn + yn = zn with x, y and z all nonzero integers, which we may assume are relatively prime. Let ζ = e2πi/n, so that ζn = 1, and consider Z[ζ] = {P(ζ), P ∈ Z[x]}, the smallest subring of C containing ζ. Then, in this ring, we have n (cid:89) xn = zn −yn = (z −ζky). k=1 Lam´e claimed that to conclude that each factor z −ζky is an nth power, it suffices to show that these factors are pairwise coprime. If this were true, then we would be able to find integers x(cid:48), y(cid:48) and z(cid:48), smaller than x,y and z but all nonzero, such that x(cid:48)n + y(cid:48)n = z(cid:48)n; this would lead to an “infinite descent” and thus prove the theorem. 4 Unfortunately, Lam´e’s claim relied on the supposition that factorisation into irreducibles is unique, and while this is true in Z, we now know that it need not be true in more general rings such as Z[ζ]. For instance, in the ring √ √ Z[ −5] = {a+b −5, a,b ∈ Z} √ √ where −5 means i 5, we have the two different factorisations √ √ 6 = 2×3 = (1+ −5)×(1− −5), where each factor is irreducible. Lam´e was thus unable to justify his claim, and the theorem remained unproved for almost 150 years. √ The numbers ζ and −5 are examples of algebraic numbers. The goal of this course is to study the property of such numbers, and of rings such that Z[ζ], so as to know what we are allowed to do with them, and what we are not. As an application, we will see how to solve certain Diophantine equations. References This module is based on the book Algebraic Number Theory and Fermat’s Last Theorem, by I.N. Stewart and D.O. Tall, published by A.K. Peters (2001). The contents of the module forms a proper subset of the material in that book. (The earlier edition, published under the title Algebraic Number Theory, is also suitable.) For alternative viewpoints, students may also like to consult the books A Brief Guide to Algebraic Number Theory, by H.P.F. Swinnerton-Dyer (LMS Student Texts # 50, CUP), or Algebraic Number Theory, by A. Fr¨ohlich and M.J. Taylor (CUP). Finally, students interested in the algorithmic side of things should consult A course in computational algebraic number theory by H. Cohen (Graduate Texts in Mathematics # 138, Springer). 5 Chapter 1 Number fields 1.1 Resultants Before we actually get started with number theory, let us introduce a tool which will turn out to be very valuable. Definition 1.1.1. Let K be a field, and let A = (cid:80)m a xj and B = j=0 j (cid:80)n b xk be two polynomials with coefficients in K. The resultant of A k=0 k and B is the (m+n)×(m+n) determinant (cid:12) (cid:12) (cid:12)a a ··· a 0 ··· 0(cid:12) m m−1 0 (cid:12)(cid:12) 0 a a ··· a ... ... (cid:12)(cid:12) (cid:12) m m−1 0 (cid:12) (cid:12)(cid:12) ... ... ... ... ... 0(cid:12)(cid:12) (cid:12) (cid:12) (cid:12) 0 ··· 0 a a ··· a (cid:12) Res(A,B) = (cid:12) m m−1 0(cid:12), (cid:12)b b ··· b 0 ··· 0(cid:12) (cid:12) n n−1 0 (cid:12) (cid:12)(cid:12) 0 b b ··· b ... ... (cid:12)(cid:12) (cid:12) n n−1 0 (cid:12) (cid:12)(cid:12) ... ... ... ... ... 0(cid:12)(cid:12) (cid:12) (cid:12) (cid:12) 0 ··· 0 bn bn−1 ··· b0(cid:12) wherethefirstnrowscontainthecoefficientsofAandthemlastonescontain those of B. The main properties of the resultant are the following: 6 Theorem 1.1.2. • Res(A,B) ∈ K, and in fact, if the coefficients of both A and B lie in a subring R of K, then Res(A,B) ∈ R. • If we can factor (over K or over a larger field) A and B as degA degB (cid:89) (cid:89) A = a (x−α ) and B = b (x−β ), j k j=1 k=1 then degA degAdegB (cid:89) (cid:89) (cid:89) Res(A,B) = adegB B(α ) = adegBbdegA (α −β ) j j k j=1 j=1 k=1 degB (cid:89) = (−1)degAdegBbdegA A(β ) = (−1)degAdegBRes(B,A). k k=1 • Res(A,B) = 0 if and only if A and B have a common factor in K[x]. Example 1.1.3. Take K = Q, A = x2 −2 ∈ Q[x] and B = x2 +1 ∈ Q[x]. Since actually A and B lie in Z[x], we have Res(A,B) ∈ Z; this is simply because by definition, (cid:12) (cid:12) (cid:12)1 0 −2 0 (cid:12) (cid:12) (cid:12) (cid:12)0 1 0 −2(cid:12) Res(A,B) = (cid:12) (cid:12). (cid:12)1 0 1 0 (cid:12) (cid:12) (cid:12) (cid:12)0 1 0 1 (cid:12) Besides, since we have √ √ A = (x− 2)(x+ 2) and B = (x−i)(x+i) over C, we find that √ √ √ √ √ √ Res(A,B) = B( 2)B(− 2) = A(i)A(−i) = ( 2−i)( 2+i)(− 2−i)(− 2+i) = 9. Example 1.1.4. Suppose we have A = BQ + R in K[x], and let b be the leading coefficient of B. Then Res(A,B) = (−1)degAdegBbdegA−degRRes(B,R). This gives a way to compute Res(A,B) by performing successive Euclid- iandivisions, whichismoreefficient(atleastforacomputer)thancomputing a large determinant when the degrees of A and B are large. 7 1.2 Field extensions 1.2.1 Notation Let K and L be fields such that K ⊆ L. One says that K is a subfield of L, and that L is an extension of K. In what follows, whenever α ∈ L (resp. α ,α ,··· ∈ L), we will write 1 2 K(α) (resp. K(α ,α ,···)) to denote the smallest subfield of L containing 1 2 K as well as α (resp. α ,α ,···). For example, we have C = R(i), and 1 2 K(α) = K if anf only if α ∈ K. Also, when R is a subring of K, we will write R[α] = {P(α), P ∈ R[x]} to denote the smallest subring of L containing R as well as α, and similarly R[α ,··· ,α ] = {P(α ,··· ,α ), P ∈ R[x ,··· ,x ]}. 1 n 1 n 1 n Example 1.2.1. The ring K[α] is a subring of the field K(α). 1.2.2 Algebraic elements, algebraic extensions Definition 1.2.2. Let α ∈ L. Then set of polynomials P ∈ K[x] such that P(α) = 0 is an ideal V of K[x], and one says that α is algebraic over K α if this ideal is nonzero, that is to say if there exists a nonzero P ∈ K[x] which vanishes at α. Else one says that α is transcendental over K, or just transcendental (for short) when K = Q. In the case when α is algebraic over K, the ideal V can be generated by α one polynomial since the ring K[x] is a PID. This polynomial is unique up to scaling, so there is a unique monic polynomial m (x) that generates V . α α This polynomial m (x) is called the minimal polynomial of α over K. One α then says that α is algebraic over K of degree n, where n = degm ∈ N, and α one writes deg α = n. When K = Q, one says for short that α is algebraic K of degree n. If every element of L is algebraic over K, one says that L is an algebraic extension of K. 8 Theorem 1.2.3. Let L/K be a field extension, and let α ∈ L be algebraic over K of degree n. Then K[α] is a field, so it agrees with K(α). It is also a vector space of dimension n over K, with basis 1,α,α2,··· ,αn−1, which we write as n−1 (cid:77) K(α) = K[α] = Kαj. j=0 Remark 1.2.4. On the other hand, if α ∈ L is transcendental over K, then it is not difficult to see that K(α) = {r(α), r ∈ K(x)} is isomorphic to the field K(x) of rational fractions over K, whence the notation K(α); in particular, it is infinite-dimensional as a K-vector space, and K[α] is a strict subring of K(α). Proof of theorem 1.2.3. Let us begin with the second equality. Let m(x) = m (x) ∈ K[x] be the minimal polynomial of α over K, an irreducible poly- α nomial of degree n. For all P(x) ∈ K[x], euclidian division in K[x] tells us that we may write P(x) = m(x)Q(x)+R(x) where Q(x),R(x) ∈ K[x] and degR(x) < n. Evaluating at x = α, we find that P(α) = R(α), so that (cid:40) (cid:41) n−1 (cid:88) K[α] = λ αj, λ ∈ K . j j j=0 Besides, if we had a relation of the form n−1 (cid:88) λ αj = 0 j j=0 with the λ in K and not all zero, this would mean that the nonzero poly- j nomial n−1 (cid:88) λ xj ∈ K[x] j j=0 9