Fact-checked by Grok 2 weeks ago

Class number formula

The class number formula, more precisely known as the analytic class number formula, is a cornerstone theorem in algebraic number theory that expresses the residue of the Dedekind zeta function of a number field at s=1 in terms of arithmetic invariants of the field, including its class number, regulator, number of roots of unity, and discriminant.^[1] For a number field K of degree n = r_1 + 2r_2 over \mathbb{Q}, where r_1 is the number of real embeddings and r_2 the number of pairs of complex embeddings, the formula states that the residue \operatorname{Res}_{s=1} \zeta_K(s) = \frac{2^{r_1} (2\pi)^{r_2} h_K R_K}{w_K \sqrt{|\Delta_K|}}, with h_K denoting the class number (the order of the ideal class group of the ring of integers \mathcal{O}_K), R_K the regulator (measuring the covolume of the unit group \mathcal{O}_K^\times in the logarithmic embedding), w_K the number of roots of unity in K, and \Delta_K the absolute discriminant of K.^[1] This relation, first established by Dirichlet for quadratic fields and generalized by Dedekind and Hecke, bridges algebraic structures like ideal classes with analytic objects such as zeta functions, highlighting the deep interplay between geometry of numbers and complex analysis in the study of number fields.^[1]^[2] In the special case of quadratic fields K = \mathbb{Q}(\sqrt{d}) with fundamental discriminant d, the formula simplifies significantly: for imaginary quadratic fields (d < 0), it becomes h_K = \frac{w_K \sqrt{|d|}}{2\pi} L(1, \chi_d), where \chi_d is the Kronecker symbol character and L(s, \chi_d) is the associated Dirichlet L-function, while for real quadratic fields (d > 0), h_K = \frac{\sqrt{d}}{2 \log \varepsilon} L(1, \chi_d) with \varepsilon the fundamental unit.^[3] These Dirichlet class number formulas not only provide explicit computations for small discriminants but also underpin results like the finiteness of the class number and effective bounds on its size, influencing topics from the distribution of primes in arithmetic progressions to the geometry of modular forms.^[3] The formula's residue interpretation arises from the functional equation of \zeta_K(s) and Poisson summation over ideals, underscoring its role in the broader arithmetic of L-functions and their special values.^[4]

Fundamentals and Historical Context

Definition and Basic Concepts

In algebraic number theory, for a number field K with ring of integers \mathcal{O}_K, the group of fractional ideals J_K consists of all nonzero fractional ideals of \mathcal{O}_K, which form an abelian group under multiplication. The principal fractional ideals P_K form a subgroup, and the ideal class group \mathrm{Cl}_K is defined as the quotient group J_K / P_K.^[5] The class number h_K is the order of \mathrm{Cl}_K, providing a measure of the failure of unique factorization in \mathcal{O}_K. Specifically, h_K = 1 if and only if every ideal in \mathcal{O}_K is principal, making \mathcal{O}_K a principal ideal domain with unique factorization up to units.^[5] The discriminant D_K of K is the determinant of the trace form on \mathcal{O}_K, an integer that encodes information about the arithmetic structure of K. It plays a key role in quantifying ramification, as the prime factors of D_K are exactly the primes that ramify in the extension K/\mathbb{Q}.^[6] The Dedekind zeta function \zeta_K(s) of K is defined for \Re(s) > 1 as the Dirichlet series \sum_{\mathfrak{a}} 1 / N(\mathfrak{a})^s, where the sum runs over all nonzero integral ideals \mathfrak{a} of \mathcal{O}_K and N(\mathfrak{a}) is the absolute norm of \mathfrak{a}. This function generalizes the Riemann zeta function and captures global arithmetic data of K.^[7] A representative example is the imaginary quadratic field K = \mathbb{Q}(i), whose ring of integers is the Gaussian integers \mathbb{Z}; here h_K = 1, so \mathbb{Z} has unique factorization into primes. In contrast, for K = \mathbb{Q}(\sqrt{-5}), the class number h_K = 2, reflecting the non-principal ideals such as (2, 1 + \sqrt{-5}).^[8]^[3] These notions underpin the class number formulas, which relate h_K to values of \zeta_K(s) and associated L-functions at s=1.^[7]

Historical Development

The development of the class number formula traces its origins to the early 19th century, beginning with Carl Friedrich Gauss's foundational work on binary quadratic forms. In his 1801 treatise Disquisitiones Arithmeticae, Gauss introduced the notion of equivalence classes of quadratic forms, demonstrating that the number of such classes—now known as the class number—for imaginary quadratic fields is finite and computing explicit values for several small discriminants, such as h(-4) = 1 and h(-8) = 1.^[9] These computations motivated deeper inquiries into the arithmetic structure of quadratic fields, laying the groundwork for analytic approaches to class numbers.^[10] A pivotal advancement came in 1837 when Peter Gustav Lejeune Dirichlet derived the first explicit analytic formula relating the class number of imaginary quadratic fields to the value of associated L-functions at s=1, employing Dirichlet characters modulo the discriminant and a partial fraction decomposition of the cotangent function.^[11] Published in Recherches sur diverses applications de l'analyse infinitésimale à la théorie des nombres, this result provided a precise connection between the class number and the arithmetic of the field, resolving conjectures posed by Gauss and establishing a bridge between algebraic and analytic number theory. During the 1850s and 1870s, Ernst Kummer and Leopold Kronecker extended these ideas through their work on ideal theory and L-functions. Kummer, in his 1844-1850 investigations into Fermat's Last Theorem, introduced "ideal numbers" to restore unique factorization in cyclotomic fields and developed the first L-functions for non-principal characters in 1849, enabling evaluations that informed class number computations in broader contexts.^[12] Kronecker, building on this in the 1870s, formulated limit formulas for L-functions associated to quadratic fields, which anticipated key residues and facilitated generalizations beyond quadratic cases.^[13] The general form of the formula for arbitrary number fields was stated by David Hilbert in his 1897 report Die Theorie der algebraischen Zahlkörper (the "Zahlbericht"), expressing the product of the class number and regulator in terms of the residue at s=1 of the Dedekind zeta function, along with the discriminant and embedding signatures; the full proof was provided by Erich Hecke in 1918.^[14] This analytic framework unified earlier results and spurred further research. In the 1930s, Carl Ludwig Siegel refined these insights with bounds on class numbers for quadratic fields; in particular, his 1935 theorem established that the class number h(D) satisfies h(D) \gg |D|^{1/2 - \epsilon} for any \epsilon > 0, providing asymptotic growth estimates tied to the class number formula despite the bounds' ineffectiveness from potential Siegel zeros.^[15]

Dirichlet Class Number Formula

Statement for Quadratic Fields

The Dirichlet class number formula provides an explicit expression for the class number of the ideal class group of a quadratic number field K = \mathbb{Q}(\sqrt{d}), where d is a square-free integer not equal to 0 or 1, in terms of the field's discriminant D, the Dirichlet L-function associated to the quadratic character \chi_D, and arithmetic invariants specific to the field.^[16] The discriminant D is defined as D = d if d \equiv 1 \pmod{4} and D = 4d otherwise, and it must be a fundamental discriminant.^[3] The quadratic character \chi_D is the Kronecker symbol (D / \cdot), and the associated Dirichlet L-function is given by

L(s, \chi_D) = \sum_{n=1}^\infty \frac{\chi_D(n)}{n^s}

for \Re(s) > 1, which admits an analytic continuation to the entire complex plane.^[16] For imaginary quadratic fields, where D < 0, the class number h is expressed as

h = \frac{w \sqrt{|D|}}{2\pi} L(1, \chi_D),

where w is the number of roots of unity in the ring of integers of K (specifically, w = 2 except for w = 4 when D = -4 and w = 6 when D = -3).^[3] This formula relates the algebraic structure of the class group directly to the special value of the L-function at s = 1.^[16] For real quadratic fields, where D > 0, the formula involves the regulator R = \log \varepsilon, with \varepsilon > 1 the fundamental unit of the ring of integers, and takes the form

h R = \frac{\sqrt{D}}{2} L(1, \chi_D),

where h is the class number (the order of the ideal class group). The narrow class number (the order of the narrow ideal class group, which accounts for principal ideals generated by elements of positive norm under both embeddings) coincides with the class number when the fundamental unit has norm -1, and is otherwise twice the class number.^[16]^[17] A notable example is the imaginary quadratic field K = \mathbb{Q}(\sqrt{-163}), which has discriminant D = -163 and class number h = 1; this is the imaginary quadratic field with the largest absolute discriminant among the exactly nine such fields with class number 1.

Analytic Proof Outline

The analytic proof of Dirichlet's class number formula for quadratic fields begins with the decomposition of the Dedekind zeta function \zeta_K(s) for a quadratic extension K = \mathbb{Q}(\sqrt{D}), where D is the discriminant. Specifically, \zeta_K(s) = \zeta(s) L(s, \chi_D), with \zeta(s) the Riemann zeta function and L(s, \chi_D) the Dirichlet L-function attached to the primitive real character \chi_D given by the Kronecker symbol (D/\cdot). This factorization arises from matching the Euler products: the primes in \mathbb{Q} factor in the ring of integers of K according to whether \chi_D(p) = 1 (split), -1 (inert), or $0 (ramified), yielding local factors that separate into those of \zeta(s) and L(s, \chi_D).^[18]^[19] The Dirichlet L-function L(s, \chi) is initially defined for \operatorname{Re}(s) > 1 by the absolutely convergent Euler product \prod_p (1 - \chi(p) p^{-s})^{-1} or the Dirichlet series \sum_{n=1}^\infty \chi(n) n^{-s}. Analytic continuation to the entire complex plane, with no poles, follows from the functional equation \Lambda(s, \chi) = |D|^{s/2} (2\pi)^{-s} \Gamma(s) L(s, \chi) = \varepsilon(\chi) \Lambda(1-s, \bar{\chi}), where \varepsilon(\chi) is the root number and \Lambda is the completed L-function; this equation is derived from the Poisson summation formula applied to Gaussian sums associated to \chi. For the non-principal real primitive character \chi_D, L(s, \chi_D) is holomorphic at s=1, and since \zeta_K(s) has a simple pole there (with the same residue order as \zeta(s)), it follows that L(1, \chi_D) is finite and non-zero.^[1]^[4] To explicitly evaluate L(1, \chi), consider the partial fraction expansion \pi \cot(\pi z) = 1/z + \sum_{n=1}^\infty \left( 1/(z-n) + 1/(z+n) \right), which has simple poles at integers with residue 1. The residue theorem applied to the contour integral \frac{1}{2\pi i} \oint_C \pi \cot(\pi z) L(1-2z, \chi) \, dz, where C is a large square contour avoiding the poles and shifted to enclose the negative real axis (leveraging the analytic continuation of L), captures the residues at z = 0, -1, -2, \dots. This yields L(1, \chi) = \sum_{n=1}^\infty \chi(n)/n, confirming convergence at the boundary \operatorname{Re}(s)=1 and providing a representation as an alternating or signed sum that is positive for real primitive \chi (due to the character's properties and the integral's evaluation). Alternatively, partial fraction decomposition directly on the series sum interchanges summation and residues to obtain the same value.^[4]^[18] The class number enters through the residue of \zeta_K(s) at s=1. The limit \lim_{s \to 1^+} (s-1) \zeta_K(s) equals this residue, which equals \lim_{s \to 1^+} (s-1) \zeta(s) L(s, \chi_D) = [L(1, \chi_D)](/page/L') since the residue of \zeta(s) is 1. On the other hand, from the ideal-theoretic definition \zeta_K(s) = \sum_{\mathfrak{a}} (\mathrm{Nm} \mathfrak{a})^{-s} (sum over nonzero ideals \mathfrak{a}), partial summation or Tauberian arguments relate the residue to the density of ideals, yielding \operatorname{Res}_{s=1} \zeta_K(s) = 2^{r_1} (2\pi)^{r_2} h R / (w \sqrt{|D|}), where h is the class number, R the regulator, w the number of roots of unity, and (r_1, r_2) the signature (either (0,1) for imaginary quadratic fields or (2,0) for real quadratic fields). Equating the two expressions for the residue gives the class number formula, specialized as h = \frac{w \sqrt{|D|} \, [L(1, \chi_D)](/page/L')}{2^{r_1} (2\pi)^{r_2} R} (with R=1 for imaginary quadratic).^[1]^[19] The non-vanishing of L(1, \chi) for real primitive characters \chi is essential, as it ensures the residue computation is valid and h > 0; this follows from the contour integral representation showing L(1, \chi) > 0 (the sum alternates positively due to \chi's sign changes) or from the functional equation implying no zero at s=1. The Generalized Riemann Hypothesis strengthens this by implying L(1, \chi) \gg |D|^\epsilon for any \epsilon > 0, with implications for bounding class numbers, though the basic non-vanishing suffices for the formula.^[18]^[4] Dirichlet's original 1837 proof employed a precursor to this analytic method, focusing on quadratic fields via partial summation on character sums.^[18]

General Class Number Formula

Statement for Arbitrary Number Fields

The analytic class number formula provides an exact relation between the class number h_K of the ring of integers \mathcal{O}_K in a number field K and the residue of its Dedekind zeta function at s=1. For a finite extension K/\mathbb{Q} of degree n = [K:\mathbb{Q}], the formula states that

h_K R_K = \frac{w_K \sqrt{|D_K|}}{2^{r_1} (2\pi)^{r_2}} \operatorname{Res}_{s=1} \zeta_K(s),

where R_K is the regulator of the unit group of K, w_K is the number of roots of unity in K, D_K is the discriminant of K, r_1 is the number of real embeddings of K, and r_2 is the number of pairs of complex conjugate embeddings (with r_1 + 2r_2 = n).^[20]^[21] The residue \operatorname{Res}_{s=1} \zeta_K(s) = \lim_{s \to 1} (s-1) \zeta_K(s) is often termed the analytic class number, as it encodes arithmetic information about K through the analytic properties of \zeta_K(s). This residue is positive and finite, reflecting the simple pole of \zeta_K(s) at s=1.^[1] In the special case of cyclotomic fields K = \mathbb{Q}(\zeta_p) for an odd prime p, the formula connects the class number to the vanishing behavior of certain L-values; specifically, if p divides the numerator of a Bernoulli number B_k for even $2 \leq k \leq p-3, then p divides h_K, rendering p an irregular prime. Effective versions of the formula yield lower bounds on h_K. By Siegel's theorem, for any \epsilon > 0, there exists a constant c(\epsilon) > 0 (ineffective) such that h_K > c(\epsilon) |D_K|^{1/2 - \epsilon} for imaginary quadratic fields, with generalizations to arbitrary number fields via the Brauer-Siegel theorem providing asymptotic growth estimates.^[22]

Key Components and Interpretations

The discriminant D_K of a number field K of degree n = [K : \mathbb{Q}] is a fundamental arithmetic invariant that encodes the ramification occurring in the extension K / \mathbb{Q}. It is defined as the ideal norm N_{K / \mathbb{Q}}(\mathfrak{D}_{K / \mathbb{Q}}), where \mathfrak{D}_{K / \mathbb{Q}} is the different ideal, given by the product \prod_{\mathfrak{p}} \mathfrak{D}_{\mathfrak{p}} over all nonzero prime ideals \mathfrak{p} of the ring of integers \mathcal{O}_K, with each local different \mathfrak{D}_{\mathfrak{p}} determined by the ramification index and residue degree at the prime below \mathfrak{p}. This structure highlights how D_K arises from local contributions at each prime, vanishing precisely when the extension is unramified everywhere. Geometrically, |D_K|^{1/2} represents the covolume of the lattice \mathcal{O}_K embedded in the Minkowski space K \otimes_{\mathbb{Q}} \mathbb{R} \cong \mathbb{R}^{r_1} \times \mathbb{C}^{r_2}, where r_1 and r_2 are the numbers of real and pairs of complex embeddings; thus, |D_K|^{1/(2n)} serves as a geometric mean measuring the average "stretching" or ramification per dimension in this embedding.^[23]^[1] The regulator R_K captures the multiplicative structure of the units in \mathcal{O}_K. By Dirichlet's unit theorem, the unit group decomposes as \mathcal{O}_K^\times \cong \mu_K \times \mathbb{Z}^{r_1 + r_2 - 1}, where \mu_K is the torsion subgroup of roots of unity. The regulator is the absolute value of the determinant of the (r_1 + r_2 - 1) \times (r_1 + r_2 - 1) matrix formed by the coordinates, under the Archimedean logarithm map \Log: K^\times \to \mathbb{R}^{r_1 + r_2}, of a \mathbb{Z}-basis of fundamental units projected onto the trace-zero hyperplane \mathbb{R}^{r_1 + r_2}_0 = \{ (x_v) \mid \sum x_v = 0 \}. This determinant quantifies the covolume of the image lattice \Log(\mathcal{O}_K^\times / \mu_K) in \mathbb{R}^{r_1 + r_2}_0, providing a measure of how densely the logarithms of units fill this space and reflecting the growth rate of units under the embeddings.^[1]^[4] The number of roots of unity w_K is the cardinality of the torsion subgroup \mu_K \subseteq \mathcal{O}_K^\times, consisting of the roots of unity in K. It accounts for the finite-order units that do not contribute to the infinite-rank part of the unit group. For instance, in real quadratic fields, w_K = 2 (just \pm 1), while in the Gaussian integers of \mathbb{Q}(i), w_K = 4 (including \pm 1, \pm i); in cyclotomic fields, it grows with the conductor. This invariant normalizes the class number by adjusting for these torsional equivalences in principal ideals.^[1]^[4] These components connect deeply to arithmetic geometry, particularly through analogies with the Birch and Swinnerton-Dyer (BSD) conjecture for elliptic curves over K. Here, the class number h_K parallels the order of the Tate-Shafarevich group \Sha(E/K), the regulator R_K mirrors the regulator of the Mordell-Weil group E(K), and w_K with factors from D_K evoke Tamagawa numbers at infinite places, linking the formula to broader conjectures on special values of L-functions and adelic measures. This perspective, rooted in the equivariant Tamagawa number conjecture, underscores the class number formula as a special case of motivic arithmetic.^[24] Overall, the formula interprets an "arithmetic density" embodied in h_K R_K / (w_K \sqrt{|D_K|})—balancing the scarcity of ideal classes, the expanse of units, and the field's ramification—against an "analytic density" from \Res_{s=1} \zeta_K(s) / (2^{r_1} (2\pi)^{r_2}), where the residue encodes global distribution of primes via the Dedekind zeta function, normalized by Archimedean contributions. This equivalence bridges the discrete world of ideals and units with the continuous analytic continuation of zeta functions, a cornerstone of modern number theory first articulated in general form by Hilbert in his 1897 Zahlbericht.^[1]^[21]^[25]

Proof Techniques

Residue Calculation via Zeta Functions

The Dedekind zeta function \zeta_K(s) for a number field K of degree n over \mathbb{Q} is defined for \Re(s) > 1 by the Dirichlet series \zeta_K(s) = \sum_{I \neq (0)} N(I)^{-s}, where the sum is over nonzero ideals I of the ring of integers \mathcal{O}_K and N(I) is the norm of I. This series admits an Euler product representation \zeta_K(s) = \prod_{\mathfrak{p}} (1 - N(\mathfrak{p})^{-s})^{-1}, where the product runs over the nonzero prime ideals \mathfrak{p} of \mathcal{O}_K.^[1] For ramified primes, the local factors take the same form, but the norms N(\mathfrak{p}) reflect the ramification index and residue degree in the decomposition of the underlying rational prime; specifically, if a rational prime p ramifies as \mathfrak{p}^e with residue degree f, then N(\mathfrak{p}) = p^f and the factor is (1 - p^{-f s})^{-1}. This structure allows approximation of \zeta_K(s) near s=1 by finite products over prime ideals of small norm, with the tail estimated using analytic properties under assumptions like the generalized Riemann hypothesis.^[26] The function \zeta_K(s) has a simple pole at s=1, arising from the volume growth of the ideal monoid under the norm map. The number of ideals with N(I) \leq x grows asymptotically as \kappa x, where \kappa = \Res_{s=1} \zeta_K(s) is the residue, reflecting the degree-n embedding of the ideals into the multiplicative group of positive rationals modulo units. Landau's Tauberian theorem provides the link: if a Dirichlet series with nonnegative coefficients converges for \Re(s) > 1, admits a meromorphic continuation to \Re(s) \geq 1 except for a simple pole at s=1 with residue \kappa, and satisfies a growth condition in the critical strip, then the partial sums of the coefficients up to x are asymptotic to \kappa x. Applied to \zeta_K(s), this yields the ideal counting function \pi_K(x) = \# \{ I : N(I) \leq x \} \sim \kappa x as x \to \infty, inverting to express the residue as the leading constant in the asymptotic.^[27] In the idelic formulation, the residue admits an expression involving a product over all places v of K (finite and infinite) and the geometry of the idele group J_K. Specifically, \kappa = \prod_v (1 - N(v)^{-1})^{-1} \cdot \covol(J_K / K^\times), where the product is a regularized Euler product over local norms at places v, and the covolume term encodes the arithmetic structure, including the class number h_K as the index [J_K : K^\times \cdot \widehat{\mathcal{O}}_K^\times] (with \widehat{\mathcal{O}}_K^\times the finite adeles of units) and the regulator R_K as the covolume of the image of \mathcal{O}_K^\times in the archimedean Lie group. This perspective ties the residue to the volume of the fundamental domain in the idelic class space, with the infinite places contributing factors like $2^{r_1} (2\pi)^{r_2} after normalization. The discriminant d_K enters via Dedekind's discriminant theorem, which expresses |d_K| as a product over ramified primes involving their ramification indices and residue degrees, appearing in the residue as \sqrt{|d_K|} in the denominator of the full arithmetic expression \kappa = 2^{r_1} (2\pi)^{r_2} h_K R_K / (w_K \sqrt{|d_K|}), where

w_K = \# \mathcal{O}_K^\times_{\tors}

is the number of roots of unity.^[1] The functional equation of \zeta_K(s) aids in the analytic continuation required for the Tauberian application but is not central to the residue computation itself.^[27] A representative numerical example is the computation of the residue for the cubic field

K = \mathbb{Q}(\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{2})

, with minimal polynomial x^3 - 2, discriminant d_K = -108, class number h_K = 1, and torsion w_K = 2. Here r_1 = 1, r_2 = 1, and the unit rank is 1 with fundamental unit \varepsilon = 1 + 2^{1/3} + 2^{2/3} (norm -1), yielding regulator R_K = \log \varepsilon \approx 1.3473. The residue is then \kappa = 2^{r_1} (2\pi)^{r_2} h_K R_K / (w_K \sqrt{|d_K|}) = 4\pi R_K / (2 \sqrt{108}) \approx 0.8146, verifiable via partial Euler products over prime ideals of norm up to a bound (e.g., X \approx 10^6) with error controlled under GRH or via databases like LMFDB.^[28]

Functional Equation and Analytic Continuation

The functional equation for the Dedekind zeta function \zeta_K(s) of a number field K is expressed in terms of the completed function \Lambda_K(s) = |D_K|^{s/2} \left( \Gamma\left(\frac{s}{2}\right) \right)^{r_1} \left( 2 (2\pi)^{-s} \Gamma(s) \right)^{r_2} \zeta_K(s), where D_K is the discriminant of K, r_1 is the number of real embeddings, and r_2 is the number of pairs of complex embeddings. This satisfies \Lambda_K(s) = \varepsilon_K \Lambda_K(1-s), where the root number \varepsilon_K = \pm 1 depends on K.^[7] The gamma factors in this equation encode the contributions from the archimedean completions of [K](/page/K): each real place contributes a factor of \Gamma(s/2), while each complex place contributes $2 (2\pi)^{-s} \Gamma(s). These factors arise from integral representations involving theta series over the adele ring, analogous to the Riemann zeta function case, and their known meromorphic properties (poles at non-positive integers) facilitate the symmetric relation across the critical line \operatorname{Re}(s) = 1/2.^[7] The functional equation enables the meromorphic continuation of \zeta_K(s) to the entire complex plane \mathbb{C}, with the only singularity being a simple pole at s=1. This continuation follows from expressing \zeta_K(s) as an Euler product over Hecke L-functions L(s, \chi) associated to grossencharacters of the ideal class group of K, each of which admits an integral representation (via Epstein zeta functions or Poisson summation) that provides meromorphic continuation and a functional equation. Since there are finitely many such characters and the product converges absolutely for \operatorname{Re}(s) > 1, the resulting \zeta_K(s) inherits these properties, with the pole at s=1 arising solely from the trivial character term.^[29] The residue of \zeta_K(s) at s=1 is positive, as established by applying Perron's formula to the Dirichlet series \sum_{\mathfrak{a}} N(\mathfrak{a})^{-s}, where \mathfrak{a} ranges over nonzero ideals of the ring of integers of K. Perron's formula yields an asymptotic \sum_{N(\mathfrak{a}) \leq x} 1 \sim \operatorname{Res}_{s=1} \zeta_K(s) \cdot x + O(x^{1-1/[K:\mathbb{Q}]}), and the left side counts ideals positively, implying the residue must be positive; this positivity is reinforced by the special values of \zeta_K(s) at negative integers being rational (hence non-negative in relevant cases) via the functional equation.^[1] The analytic continuation and functional equation for general number fields were established by Erich Hecke in the early 1920s, building on Dedekind's definition and extending Dirichlet's work for quadratic fields.^[1]

Extensions and Generalizations

Formulas for Galois Extensions

In Galois extensions K/\mathbb{Q} of number fields, where G = \mathrm{Gal}(K/\mathbb{Q}) is finite, the Dedekind zeta function \zeta_K(s) decomposes as a product of Artin L-functions associated to the irreducible characters of G:

\zeta_K(s) = \prod_{\chi \in \widehat{G}} L(s, \chi)^{\chi(1)},

where \widehat{G} denotes the set of irreducible characters \chi of G, \chi(1) is the degree of the representation affording \chi, and L(s, \chi) is the Artin L-function.^[30] This factorization reflects the action of the Galois group on the ideals of the ring of integers of K.^[4] The residue of \zeta_K(s) at s=1 equals \operatorname{Res}_{s=1} \zeta(s) \prod_{\chi \neq 1} L(1, \chi)^{\chi(1)}, assuming the holomorphy of L(s, \chi) at s=1 for non-trivial \chi (as conjectured by Artin and verified in many cases). Using the analytic class number formula, this yields

h_K = \frac{ w_K \sqrt{|\Delta_K|} \cdot \left( \frac{\pi^2}{6} \prod_{\chi \neq 1} L(1, \chi)^{\chi(1)} \right) }{ 2^{r_1} (2\pi)^{r_2} R_K },

where r_1 and r_2 are the numbers of real and complex embeddings, w_K is the number of roots of unity, \Delta_K is the discriminant, and R_K is the regulator of the unit group.^[4] Here, the Artin conductor enters implicitly through the definition of L(s, \chi), which incorporates ramification data via the conductor-discriminant formula for each \chi.^[30] The regulator R_K accounts for the Galois action on the units, often computed as the determinant of the logarithm map on a basis respecting Galois orbits, ensuring equivariance under G.^[4] A specific case arises for cyclotomic fields K = \mathbb{Q}(\zeta_n), which are abelian Galois extensions with [G](/page/G) \cong (\mathbb{Z}/n\mathbb{Z})^\times. The zeta function decomposes as \zeta_K(s) = \prod_{\chi \bmod n} L(s, \chi), and the class number formula expresses h_K in terms of the product \prod_{\chi \neq 1} L(1, \chi) via the residue at s=1, incorporating the regulator and discriminant specific to cyclotomic fields.^[31] This links to Vandiver's conjecture, which posits that for prime [p](/page/P′′), p does not divide the class number h^+ of the maximal real subfield \mathbb{Q}(\zeta_p)^+, implying no p-torsion in the relevant L(1, \chi) factors.^[32] For dihedral Galois quartic fields (degree 4 over \mathbb{Q} with Galois group the dihedral group of order 8), the class number h_K is explicitly related to those of its three quadratic subfields F_1, F_2, F_3 via Galois cohomology and unit indices: typically h_K = \frac{h_{F_1} h_{F_2} h_{F_3}}{2} or adjusted by a power of 2 depending on ramification and unit ranks, as derived from the transfer map in the class group.^[33] This reflects the non-abelian structure, where ideals in subfields lift without splitting under the full Galois action. Unlike non-Galois extensions, where ideal classes may split arbitrarily upon embedding into the Galois closure, the Galois-invariant nature ensures that the ideal class group of K forms a module over \mathbb{Z}[G], preventing such splitting and tying h_K directly to character values without additional decomposition factors.^[4] The abelian subcase simplifies to Dirichlet L-functions but shares the product structure over characters.^[31]

Formulas for Abelian Extensions

In abelian extensions K/\mathbb{Q}, the Dedekind zeta function factors as \zeta_K(s) = \prod_{\chi \mod f} L(s, \chi), where the product runs over all Hecke characters \chi of finite order modulo the conductor f of K, and class field theory identifies the ray class group \mathrm{Cl}_f with the Galois group \mathrm{Gal}(K/\mathbb{Q}).^[34] This decomposition arises because the irreducible representations of the abelian Galois group are one-dimensional, corresponding to these Hecke characters.^[31] The residue of \zeta_K(s) at s=1 equals \operatorname{Res}_{s=1} \zeta(s) \prod_{\chi \neq \chi_0} L(1, \chi), where \chi_0 is the trivial character contributing the simple pole, allowing the class number formula to express the class number h_K and regulator R_K in terms of these L-values. For a generalized version involving a modulus m, the ray class number h_K^{(m)} of the ray class group \mathrm{Cl}_m satisfies

h_K^{(m)} R_K = \frac{ w_K \sqrt{|D_K|} \cdot \frac{\pi^2}{6} \prod_{\chi \neq \chi_0} L(1, \chi) }{ 2^{r_1} (2\pi)^{r_2} },

where the product is over all grossencharacters (Hecke characters) \chi of finite order modulo m excluding the trivial one, w_K is the number of roots of unity in K, D_K is the discriminant, and r_1, r_2 are the numbers of real and pairs of complex embeddings.^[4] This extends the standard class number formula by incorporating ray class variants, capturing arithmetic in ray class fields ramified only at primes dividing m.^[34] An idelic reformulation expresses the residue at s=1 of \zeta_K(s) as a Tamagawa measure on the adele class group A_K^\times / K^\times, where the Tamagawa number \tau_K = 1 for number fields, reflecting the volume of the fundamental domain under the Haar measure normalized appropriately.^[34] This perspective unifies local and global aspects via ideles and highlights the role of units and class groups in the measure. For example, in real abelian extensions such as the maximal real subfield \mathbb{Q}(\zeta_p + \zeta_p^{-1}) of the p-th cyclotomic field (with p prime), the class number relates to the Stickelberger ideal in the group ring \mathbb{Z}[\mathrm{Gal}(K/\mathbb{Q})], which annihilates the p-part of the class group via Stickelberger's theorem, providing explicit relations for computing or bounding the class number.^[35] These formulas enable applications to the equidistribution of class groups in families of abelian extensions, as studied through generalizations of the Cohen-Lenstra heuristics, which predict probabilities for the structure of class groups based on L-value distributions.

References

[1]
[PDF] 19 The analytic class number formula
Nov 13, 2017 · We are now ready to prove the analytic class number formula. Theorem 19.12 (Analytic Class Number Formula). Let K be a number field of degree n.
[2]
[PDF] explicit mertens' theorems for number fields and dedekind zeta ...
analytic class number formula asserts that the residue of ζK(s) at s = 1 is. κK = 2r1 (2π)r2 hKRK. wK. √. ∆K. ,. (4) where r1 is the number of real places of K ...<|control11|><|separator|>
[3]
[PDF] The ideal class number formula for an imaginary quadratic field
Our goal is a formula for h(F) to complement the algorithm that we now have for it. The formula requires elements of analytic number theory, to be presented in ...
[4]
[PDF] The Analytic Class Number Formula and L-functions - Berkeley Math
This gives an interpre- tation of the denominators of Bernoulli numbers in terms of the torsion of the algebraic K -theory. ... lytic class number formula ...
[5]
[PDF] ALGEBRAIC NUMBER THEORY Contents Introduction ...
We define the ideal class group Cl(A) of A to be the quotient Cl(A) = Id(A)/P(A) of Id(A) by the subgroup of principal ideals. The class number of A is the ...
[6]
[PDF] Discriminants and ramified primes - Keith Conrad
The discriminant of OK is discZ(OK). The next lemma says reduction modulo p commutes (in a suitable sense) with the formation of discriminants. Lemma 3.3.
[7]
[PDF] The Dedekind Zeta Function and the Class Number Formula Math ...
Definition 2.1. The Dedekind zeta function of a number field K is defined by. ζK(s) = X a. 1. N(a)s with the sum running over all the integral ideals of K. 1 ...
[8]
[PDF] 6 Gaussian Integers and Rings of Algebraic Integers
The class number is 1 precisely when Z[ζp] has unique factorization. Using these techniques, Kum- mer proved Fermat's Last Theorem for all regular primes (and ...
[9]
GAUSS' CLASS NUMBER PROBLEM FOR IMAGINARY ...
We shall also write (a, b9 c) ~ (A, B, C) if the binary quadratic form. (A9 B, C) can be obtained from (a, b9 c) by a linear change in variables (3); and we ...
[10]
[PDF] Class Numbers of Quadratic Fields
The concept of class number first occurs in Gauss's Disquisitiones Arithmeticae written in 1801. In this work, we find the beginnings of modern number theory.
[11]
Recherches sur diverses applications de l'Analyse infinitesimale à la ...
Recherches sur diverses applications de l'Analyse infinitesimale à la théorie des Nombres. · Volume: 19, page 324-369 · ISSN: 0075-4102; 1435-5345/e ...Missing: 1837 | Show results with:1837
[12]
[PDF] Kummer's theory on ideal numbers and Fermat's Last Theorem
This paper is an exposition on Ernst Kummer's theory of ideal numbers, which “saves” unique factorization in the ring of integers of the cy- clotomic field. A ...
[13]
[PDF] Kronecker's first limit formula, revisited - ETH Zürich
Perhaps the most important applications of Kronecker's limit formulas are to L-functions associated to quadratic fields and their abelian extensions. The first ...
[14]
[PDF] Takagi's Class Field Theory - RIMS, Kyoto University
ibid.) In 1897, Hilbert wrote his report [Hi‐1897] on algebraic numbers. This became a standard text for the theory of algebraic numbers. Then in the next ...
[15]
[PDF] Class Numbers of Quadratic Fields - HAL
May 1, 2020 · Perhaps the most significant of these developments is the 1935 theorem of Siegel that states that for any ϵ > 0, there is a constant C(ϵ) ...
[16]
[PDF] Algebraic Number Theory - UCSB Math
It becomes clear from his Preface that Number Theory was Neukirch's favorite subject in mathematics. He was enthusiastic about it, and he was also.
[17]
[PDF] Upper bounds on class numbers of real quadratic fields - arXiv
Jun 26, 2025 · At this point by the class number formula (1.1), the bound on the L-function. (2.9) and the bound on the regulator (2.17), we obtain. (2.18) h ...
[18]
[PDF] Dirichlet's Class Number Formula
Nov 13, 2012 · It works especially well for imaginary quadratic fields. In this problem p will be an odd prime. Let F = Q(. √. −p) and let R be the ...Missing: 1837 | Show results with:1837
[19]
[PDF] DIRICHLET L-FUNCTIONS AND DEDEKIND ζ-FUNCTIONS
Aug 30, 2016 · In section 9, we then define the Dedekind zeta function, describe the ideal class group and then highlight the Dedekind zeta function's role ...
[20]
[PDF] Algebraic Number Theory
There is one neologism that I propose in this translation, with Krgen. Neukirch's blessing: I call replete divisor, ideal, etc., what is usually called.
[21]
Analytic class number formula (reviewed) - LMFDB
The analytic class number formula gives the residue at this pole: lim ⁡ s → 1 ( s − 1 ) ζ K ( s ) = 2 r 1 ⋅ ( 2 π ) r 2 ⋅ R ⋅ h w ⋅ ∣ D ∣ . ... s→1lim (s−1)ζK(s) ...
[22]
[PDF] the different ideal - keith conrad
The prime factors of the discriminant of K are the prime numbers p that ramify in OK. Here are two related tasks that are not answered just from knowing the ...
[23]
[PDF] The Equivariant Tamagawa Number Conjecture: A survey
The Tamagawa number conjecture of Bloch and Kato [10] is a beautiful gener- alization of the analytic class number formula (this is a theorem!) on the one hand,.Missing: Sha | Show results with:Sha<|separator|>
[24]
[PDF] The Theory of Algebraic - Number Fields
Jun 14, 2015 · Gauss sum; its treatment in Hilbert's Zahlbericht is remarkable in two ways: first, Hilbert does not state (let alone prove) Stickelberger's ...
[25]
None
### Summary of Residue Computation of Dedekind Zeta Function (arXiv:1305.0035)
[26]
[PDF] introduction to zeta and l-functions from
We give two examples of so-called tauberian theorems, each one leading to a proof of the prime number theorem. ... The residue of Dedekind zeta function. The ...
[27]
[PDF] class group calculations - Keith Conrad
Example 2.3. Let K = Q( √ −14). We will show the class group is cyclic of order 4. Here n = 2,r2 = 1, and disc(K) = −56.
[28]
[PDF] Daniel A. Marcus - Number Fields
7 The Dedekind Zeta Function and the Class Number Formula ..... 129. The ... Hilbert's formula. See exercise 27.) Conclude that G is cyclic. 35. Now ...
[29]
[PDF] Hecke L-functions on Algebraic Number Fields
In the final section, I include the theorem, proven by both Hecke and. Tate, that Hecke L-functions admit meromorphic continuations to the entire complex plane.
[30]
[PDF] Artin L-Functions
Let L/K be a Galois extension of number fields with G = Gal(L/K). Then. 1. For characters χ, χ0 of G, we have f(χ + χ0) = f(χ)f(χ0). 2. If χ1 is the trivial ...
[31]
[PDF] 18 The analytic class number formula
Nov 12, 2015 · ... cyclotomic fields, but need not be cyclotomic), and for this purpose ... Let K be a number field of degree n = r + 2s with r real and s complex.
[32]
[PDF] Vandiver's Conjecture via K-theory - School of Mathematics, TIFR
Conjecture 1 (Vandiver's Conjecture) Let h+ denote the class number of the maximal totally real subfield Q(ζp)+ of Q(ζp). Then p 6 h+.
[33]
[PDF] Class groups of dihedral extensions
Let F be a number field with class number not divisible by p, L/F a dihedral extension with Galois group Gal(L/F) ≃ Dp, k its quadratic subfield, and K the.
[34]
[PDF] Class Field Theory
Class field theory describes the abelian extensions of a local or global field in terms of the arithmetic of the field itself.
[35]
[PDF] A short basis of the Stickelberger ideal of a cyclotomic field - arXiv
Sep 27, 2021 · As a direct practical consequence, we deduce from this short basis an explicit upper bound on the relative class number, that is valid for any.