Discriminant of an algebraic number field

Let K be an algebraic number field, and let \mathcal{O}_K be its ring of integers. Let b_1,\dots,b_n be an integral basis of \mathcal{O}_K (i.e. a basis as a \Z-module), and let \{\sigma_1,\dots,\sigma_n\} be the set of embeddings of K into the complex numbers (i.e. injective ring homomorphisms K\to\C). The discriminant of K is the square of the determinant of the n\times n matrix B whose (i,j)-entry is \sigma_i(b_j). Symbolically, :\Delta_K=\det\left(\begin{array}{cccc} \sigma_1(b_1) & \sigma_1(b_2) &\cdots & \sigma_1(b_n) \\ \sigma_2(b_1) & \ddots & & \vdots \\ \vdots & & \ddots & \vdots \\ \sigma_n(b_1) & \cdots & \cdots & \sigma_n(b_n) \end{array}\right)^2. Equivalently, the trace from K to \Q can be used. Specifically, define the trace form to be the matrix whose (i,j)-entry is \operatorname{Tr}_{K/\Q}(b_ib_j). This matrix equals B^T\!B, so defining the discriminant of K as the determinant of this matrix gives us the same result as the preceding definition. The discriminant of an order in K with integral basis b_1,\dots,b_n is defined in the same way. ==Examples==

• Quadratic number fields: let d be a square-free integer; then the discriminant of K=\Q(\sqrt{d}) is • Cyclotomic fields: let n>2 be an integer, let \zeta_n be a primitive n-th root of unity, and let K_n=\Q(\zeta_n) be the n-th cyclotomic field. The discriminant of K_n is given by :: \Delta_{K_n} = (-1)^{\varphi(n)/2} \frac{n^{\varphi(n)}}{\displaystyle\prod_{p|n} p^{\varphi(n)/(p-1)}} : where \varphi(n) is Euler's totient function, and the product in the denominator is over primes p dividing n. • Power bases: In the case where the ring \mathcal{O}_K of algebraic integers has a power integral basis, that is, can be written as \mathcal{O}_K=\Z[\alpha], the discriminant of K is equal to the discriminant of the minimal polynomial of \alpha. To see this, one can choose the integral basis of \mathcal{O}_K to be ::b_1=1, b_2=\alpha, b_3=\alpha^2, \dots,b_n = \alpha^{n-1}. :Then, the matrix B in the definition is the Vandermonde matrix associated to \alpha_i=\sigma_i(\alpha), whose squared determinant is :: \prod_{1\leq i, :which is exactly the definition of the discriminant of the minimal polynomial. • Let K=\Q(\alpha) be the number field obtained by adjoining a root \alpha of the polynomial x^3-x^2-2x-8. This is Richard Dedekind's original example of a number field whose ring of integers does not possess a power basis. An integral basis is given by ::\left\{1,\alpha,\frac{\alpha(\alpha+1)}{2}\right\} :and the discriminant of K is −503. • Repeated discriminants: the discriminant of a quadratic field uniquely identifies it, but this is not true, in general, for higher-degree number fields. For example, there are two non-isomorphic cubic fields of discriminant 3969. They are obtained by adjoining a root of the polynomial x^3-21x+28 or x^3-21x-35, respectively. 3 − 11x2 + x + 1. This is an example that does not have a power basis. An integral basis is given by {1, α, 1/2(α2 + 1)}, and the trace form is :: \left(\begin{array}{ccc} 3 & 11 & 61 \\ 11 & 119 & 653 \\ 61 & 653 & 3589 \\ \end{array}\right). : The discriminant of K is the determinant of this matrix, which is 1304 = 23 163. --> ==Basic results==

• '''Brill's theorem''': The sign of the discriminant is (-1)^{r_2} where r_2 is the number of complex places of K. • A prime p ramifies in K if and only if p divides \Delta_K. • '''Stickelberger's theorem''': :: \Delta_K\equiv 0\text{ or }1 \pmod 4. • '''Minkowski's bound''': Let n denote the degree of the extension K/\Q and r_2 the number of complex places of K, then :: |\Delta_K|^{1/2}\geq \frac{n^n}{n!}\left(\frac{\pi}{4}\right)^{r_2} \geq \frac{n^n}{n!}\left(\frac{\pi}{4}\right)^{n/2}. • '''Minkowski's theorem''': If K is not \Q, then |\Delta_K|>1 (this follows directly from the Minkowski bound). • Hermite–Minkowski theorem: Let N be a positive integer. There are only finitely many (up to isomorphisms) algebraic number fields K with |\Delta_K|. Again, this follows from the Minkowski bound together with Hermite's theorem (that there are only finitely many algebraic number fields with prescribed discriminant). ==History==

showed that every number field possesses an integral basis, allowing him to define the discriminant of an arbitrary number field. The definition of the discriminant of a general algebraic number field, K, was given by Dedekind in 1871. Hermite's theorem predates the general definition of the discriminant with Charles Hermite publishing a proof of it in 1857. In 1877, Alexander von Brill determined the sign of the discriminant. Leopold Kronecker first stated Minkowski's theorem in 1882, though the first proof was given by Hermann Minkowski in 1891. In the same year, Minkowski published his bound on the discriminant. Near the end of the nineteenth century, Ludwig Stickelberger obtained his theorem on the residue of the discriminant modulo four. ==Relative discriminant==

The discriminant defined above is sometimes referred to as the absolute discriminant of K to distinguish it from the relative discriminant ΔK/L of an extension of number fields K/L, which is an ideal in OL. The relative discriminant is defined in a fashion similar to the absolute discriminant, but must take into account that ideals in OL may not be principal and that there may not be an OL basis of OK. Let {σ1, ..., σn} be the set of embeddings of K into C which are the identity on L. If b1, ..., bn is any basis of K over L, let d(b1, ..., bn) be the square of the determinant of the n by n matrix whose (i,j)-entry is σi(bj). Then, the relative discriminant of K/L is the ideal generated by the d(b1, ..., bn) as {b1, ..., bn} varies over all integral bases of K/L. (i.e. bases with the property that bi ∈ OK for all i.) Alternatively, the relative discriminant of K/L is the norm of the different of K/L. When L = Q, the relative discriminant ΔK/Q is the principal ideal of Z generated by the absolute discriminant ΔK . In a tower of fields K/L/F the relative discriminants are related by :\Delta_{K/F} = \mathcal{N}_{L/F}\left({\Delta_{K/L}}\right) \Delta_{L/F}^{[K:L]} where \mathcal{N} denotes relative norm. Ramification The relative discriminant regulates the ramification data of the field extension K/L. A prime ideal p of L ramifies in K if, and only if, it divides the relative discriminant ΔK/L. An extension is unramified if, and only if, the discriminant is the unit ideal. The Minkowski bound above shows that there are no non-trivial unramified extensions of Q. Fields larger than Q may have unramified extensions: for example, for any field with class number greater than one, its Hilbert class field is a non-trivial unramified extension. ==Root discriminant==

The root discriminant of a degree n number field K is defined by the formula :\operatorname{rd}_K = |\Delta_K|^{1/n}. The relation between relative discriminants in a tower of fields shows that the root discriminant does not change in an unramified extension. Asymptotic lower bounds Given nonnegative rational numbers ρ and σ, not both 0, and a positive integer n such that the pair (r,2s) = (ρn,σn) is in Z × 2Z, let αn(ρ, σ) be the infimum of rdK as K ranges over degree n number fields with r real embeddings and 2s complex embeddings, and let α(ρ, σ) =  liminfn→∞ αn(ρ, σ). Then : \alpha(\rho,\sigma) \ge 60.8^\rho 22.3^\sigma , and the generalized Riemann hypothesis implies the stronger bound : \alpha(\rho,\sigma) \ge 215.3^\rho 44.7^\sigma . There is also a lower bound that holds in all degrees, not just asymptotically: For totally real fields, the root discriminant is > 14, with 1229 exceptions. Asymptotic upper bounds On the other hand, the existence of an infinite class field tower can give upper bounds on the values of α(ρ, σ). For example, the infinite class field tower over Q() with m = 3·5·7·11·19 produces fields of arbitrarily large degree with root discriminant 2 ≈ 296.276, ==Relation to other quantities==

• When embedded into K\otimes_\mathbf{Q}\mathbf{R}, the volume of the fundamental domain of OK is \sqrt (sometimes a different measure is used and the volume obtained is 2^{-r_2}\sqrt, where r2 is the number of complex places of K). • Due to its appearance in this volume, the discriminant also appears in the functional equation of the Dedekind zeta function of K, and hence in the analytic class number formula, and the Brauer–Siegel theorem. • The relative discriminant of K/L is the Artin conductor of the regular representation of the Galois group of K/L. This provides a relation to the Artin conductors of the characters of the Galois group of K/L, called the conductor-discriminant formula. ==Notes==