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The Ideal Class Number Formula for Imaginary Quadratic Fields

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The Ideal Class Number Formula for Imaginary Quadratic Fields THE DIRICHLET CLASS NUMBER FORMULA FOR IMAGINARY QUADRATIC FIELDS The factorizations p p 6 = 2 · 3 = (1 + −5)(1 − −5) show that unique factorization fails in the ring p p Z[ −5] = fa + b −5 : a; b 2 Zg; p because 2, 3, and 1 ± −5 are irreducible and nonassociate. These notes present a formula that in somep sense measures the extent to which unique factorization fails in settings such as Z[ −5]. The large-scale methodology deserves immediate note, before the reader is immersed in a long succession of smaller attention-demanding specifics: • algebra lets us define a group that measures the failure of unique factoriza- tion, • geometry shows that the group is finite and gives an algorithm to find a set of group element representatives in any specific instance, • and analysis yields the formula for the group's order. To move forward through the main storyline without bogging down, the expo- sition quotes results from algebra and complex analysis even though elementary arguments are possible in this context. For a more fleshed out and elementary pre- sentation, see Tom Weston's online notes for the 2004 Ross mathematics program, www.math.umass.edu/~weston/oldpapers/cnf.pdf The class number formula in general is discussed in many number theory books, such as the books by Marcus and by Borevich and Shafarevich. Contents Part 1. ALGEBRA: QUADRATIC NUMBER FIELDS 2 1. Quadratic Fields and Their Integers 2 2. The Units of a Quadratic Field 4 3. The Ideals of a Quadratic Field 5 4. Unique Factorization of Ideals 7 5. The Character of a Quadratic Field 8 6. Decomposition of Rational Primes 11 7. Fractional Ideals and the Ideal Class Group 13 8. Abelian Group Structure of Ideals 14 Part 2. GEOMETRY: COMPLEX LATTICES 15 9. Complex Lattices and Homothety 15 10. Complex Multiplication 17 1 2 THE DIRICHLET CLASS NUMBER FORMULA FOR IMAGINARY QUADRATIC FIELDS Part 3. ANALYSIS: ZETA AND L-FUNCTIONS OF AN IMAGINARY QUADRATIC FIELD 19 11. Summation by Parts and Dirichlet Series Convergence 20 12. The Euler{Riemann Zeta Function 21 13. The L-Function of a Quadratic Field 22 14. The Dedekind Zeta Function of a Quadratic Field 25 15. The Class Number Formula 29 Part 1. ALGEBRA: QUADRATIC NUMBER FIELDS p This part of these notes discusses quadraticp number fields (fields like Q( −5)) and their rings of integers (rings like Z[ −5]). The ideals of the ring factor uniquely even though the elements of the ring may not. A group called the ideal class group measures the extent to which ideals fail to correspond to ring elements, thus measuring the extent to which unique factorization of elements fails. 1. Quadratic Fields and Their Integers Definition 1.1. A quadratic number field is a field F (inside C) such that F has dimension 2 as a vector space over Q. Such a field takes the form p p F = Q( n) = fa + b n : a; b 2 Qg; n 2 Z − f0; 1g squarefree: If n is positive then F is a real quadratic number field, and if n is negative then F is an imaginary quadratic number field. From now on in this writeup the symbol F denotes a quadratic number field, and quadratic number field is freely shortened to quadratic field. The conjugation function of F is p p : F −! F; a + b n = a − b n: Conjugation is a ring homomorphism, meaning that x + y = x + y and xy = x y for all x; y 2 F: And conjugation is an involution, meaning that x = x for all x 2 F: Thus conjugation is an automorphism of F . The only other automorphism of F is the identity map, and so the group of automorphisms of F has order 2, generated by conjugation. The trace function of F is the additive homomorphism tr : F −! Q; tr(α) = α + α: Specifically, p p p tr(a + b n) = a + b n + a + b n = 2a: The norm function of F is the multiplicative homomorphism × × N: F −! Q ; N(α) = α α: Specifically, p p p N(a + b n) = (a + b n)(a + b n) = a2 − b2n: THE DIRICHLET CLASS NUMBER FORMULA FOR IMAGINARY QUADRATIC FIELDS 3 If F is imaginary quadratic then the norm is positive on F ×. Because conjugation is an involution, it has no effect on trace and norm, i.e., tr(α) = tr(α) for all α 2 F and N(α) = N(α) for all α 2 F ×. Also because conjugation is an involution, tr(α) = tr(α) and N(α) = N(α). Thus each element α of F satisfies a monic quadratic polynomial over Q, (X − α)(X − α) = X2 − tr(α)X + N(α): p This polynomial has discriminant is 4b2n where α = a+b n, and so it is irreducible exactly when α2 = Q. Of course, each α 2 Q satisfies the monic linear polynomial X − α over Q. Definition 1.2. An element of F is an integer if its minimal monic polynomial over Q in fact has coefficients in Z. Thusp the integers of F \ Q (the rational integers of F ) are Z. An element α = a + b n of F − Q is an algebraic integer if and only if its trace 2a and its norm a2 −b2n are rational integers. Supposing that 2a; a2 −b2n 2 Z, we obtain necessary conditions on a and b. The relation 4N(α) = (2a)2 − 4b2n shows that 4b2n 2 Z; if a 2 Z then it further shows that 4b2n 2 4Z, so that b 2 Z because n is squarefree; 1 2 if a2 = Z then a 2 2 Z − Z, and the relation further shows that 4b n ≡ 1 (mod 4), so 1 2 that b 2 2 Z − Z and then n ≡ 1 (mod 4) because 4b ≡ 1 (mod 4). Now we show that the necessary conditions on a and b are also sufficient. If a; b 2 Z then certainly 2 2 2a; a − b n 2 Z, and if n ≡ 1 (mod 4) and a = ao=2; b = bo=2 with ao; bo 2 Z odd 2 2 2 2 then 2a = ao 2 Z and a − b n = (ao − bon)=4 2 Z. We have established p Proposition 1.3. The integers of the quadratic field F = Q( n) are p ( 1+ n 2 if n = 1 (mod 4) OF = Z[g]; g = p n if n = 2; 3 (mod 4): The integers of F form a ring. The minimal monic polynomial in Z[X] satisfied by the OF generator g in the previous proposition is quadratic, ( X2 − X − n−1 if n = 1 (mod 4) f(X) = 4 X2 − n if n = 2; 3 (mod 4): Thus, as an abelian group the integer ring is in fact OF = gZ ⊕ Z: The discriminant of the quadratic polynomial f(X) (the quantity b2 − 4ac that goes under the square root in the quadratic formula) is n if n = 1 (mod 4) and 4n if n = 2; 3 (mod 4). This quantity, an invariant of the quadratic field F , plays a significant role in the structure of F ; in the present writeup it will manifest itself in the class number formula. Definition 1.4. The discriminant of F is ( n if n = 1 (mod 4) DF = 4n if n = 2; 3 (mod 4): 4 THE DIRICHLET CLASS NUMBER FORMULA FOR IMAGINARY QUADRATIC FIELDS The cases built into the definition of the discriminant allow it to give a uniform description of the integers, p D + D O = [r]; r = F F ; F Z 2 and similarly we will see that the discriminant gives uniform descriptions of various phenomena associated with F . The minimal polynomial of r is 2 f(X) = X − DF X + DF (DF − 1)=4; whose discriminant is indeed DF . One can think of the casewise formula for the discriminant as the result of a calculation rather than as a definition. Other definitions of the discriminant are case-free in terms of g (where OF = gZ ⊕ Z as before), although g itself involves cases, 1 g D = (det )2 F 1 g and tr(1 · 1) tr(1 · g) D = det : F tr(g · 1) tr(g · g) 2. The Units of a Quadratic Field Definition 2.1. A unit of F is an invertible element of the integer ring OF . The × unit group of F is the multiplicative group OF . Proposition 2.2. An element α of OF is a unit if and only if N(α) = ±1. Proof. If α 2 OF is multiplicatively invertible by β 2 OF then 1 = N(1) = N(αβ) = N(α)N(β); so that N(α) = ±1 because both norms are integers. Conversely, if N(α) = ±1 then α is invertible by ±α 2 OF because ±αα = ±N(α) = 1. p If K = Q( n) is imaginary quadratic then all norms a2 − b2n are nonnegative, and inspection shows that the unit group is 8 {±1; ±ig if n = −1 <> p × 2 −1+ −3 OF = {±1; ±ζ3; ±ζ3 g if n = −3 (where ζ3 = 2 ) :>{±1g otherwise: Inspection quickly shows that these groups are cyclic, but also we know that any finite subgroup of the multiplicativep group of any field is cyclic, so the inspection is only a confirmation. If K = Q( n) is real quadratic then nontrivially there exists × a so-called fundamental unit u > 1 in OF such that the unit group is × n OF = {±u : n 2 Zg ≈ Z=2Z × Z: Finding the fundamental unit amounts to solving Pell's Equation, x2 − ny2 = 1.
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