Discriminant of an algebraic number field

"Brill's theorem" redirects here. For the result in algebraic geometry, see Brill–Noether theorem.

File:Discriminant49CubicFieldFundamentalDomain.png

A fundamental domain of the ring of integers of the field K obtained from Q by adjoining a root of x³ − x² − 2x + 1. This fundamental domain sits inside K ⊗_QR. The discriminant of K is 49 = 7². Accordingly, the volume of the fundamental domain is 7 and K is only ramified at 7.

In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field. More specifically, it is proportional to the volume of the fundamental domain of the ring of integers, and it regulates which primes are ramified.

The discriminant is one of the most basic invariants of a number field, and occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K, and the analytic class number formula for K. An old theorem of Hermite states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an open problem, and the subject of current research.

The discriminant of K can be referred to as the absolute discriminant of K to distinguish it from the relative discriminant of an extension K/L of number fields. The latter is an ideal in the ring of integers of L, and like the absolute discriminant it indicates which primes are ramified in K/L. It is a generalization of the absolute discriminant allowing for L to be bigger than Q; in fact, when L = Q, the relative discriminant of K/Q is the principal ideal of Z generated by the absolute discriminant of K.

Definition

Let K be an algebraic number field, and let O_K be its ring of integers. Let b₁, ..., b_n be an integral basis of O_K (i.e. a basis as a Z-module), and let {σ₁, ..., σ_n} be the set of embeddings of K into the complex numbers (i.e. injective ring homomorphisms K → C). The discriminant of K is the square of the determinant of the n by n matrix B whose (i,j)-entry is σ_i(b_j). Symbolically,

$\Delta_K=\left(\operatorname{det}\left(\begin{array}{cccc} \sigma_1(b_1) & \sigma_1(b_2) &\cdots & \sigma_1(b_n) \\ \sigma_2(b_1) & \ddots & & \vdots \\ \vdots & & \ddots & \vdots \\ \sigma_n(b_1) & \cdots & \cdots & \sigma_n(b_n) \end{array}\right)\right)^2.$

Equivalently, the trace from K to Q can be used. Specifically, define the trace form to be the matrix whose (i,j)-entry is Tr_K/Q(b_ib_j). This matrix equals B^TB, so the discriminant of K is the determinant of this matrix.

Examples

• Quadratic number fields: let d be a square-free integer, then the discriminant of $K=\mathbf{Q}(\sqrt{d})$ is

$\Delta_K=\left\{\begin{array}{ll} d &\text{if }d\equiv 1\pmod 4 \\ 4d &\text{if }d\equiv 2,3\pmod 4. \\\end{array}\right.$

An integer that occurs as the discriminant of a quadratic number field is called a fundamental discriminant.

• Cyclotomic fields: let n > 2 be an integer, let ζ_n be a primitive nth root of unity, and let K_n = Q(ζ_n) be the nth 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 of integers has a power integral basis, that is, can be written as O_K = Z[α], the discriminant of K is equal to the discriminant of the minimal polynomial of α. To see this, one can chose the integral basis of O_K to be b₁ = 1, b₂ = α, b₃ = α², ..., b_n = αⁿ⁻¹. Then, the matrix in the definition is the Vandermonde matrix associated to α_i = σ_i(α), whose determinant squared is

$\prod_{1\leq i<j\leq n}(\alpha_i-\alpha_j)^2$

which is exactly the definition of the discriminant of the minimal polynomial.

• Let K = Q(α) be the number field obtained by adjoining a root α of the polynomial x³ − x² − 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 {1, α, α(α + 1)/2} 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 x³ − 21x + 28 or x³ − 21x − 35, respectively.

Basic results

• Brill's theorem: The sign of the discriminant is (−1)^r₂ where r₂ is the number of complex places of K.
• A prime p ramifies in K if, and only if, p divides Δ_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 r₂ 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 |Δ_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 |Δ_K| < N. Again, this follows from the Minkowski bound together Hermite's theorem (There are only finitely many algebraic number fields with prescribed discriminant).

History

Richard Dedekind 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. At this point, he already knew the relationship between the discriminant and ramification.

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 Δ_K/L of an extension of number fields K/L, which is an ideal in O_L. The relative discriminant is defined in a fashion similar to the absolute discriminant, but must take into account that ideals in O_L may not be principal and that there may not be an O_L basis of O_K. Let {σ₁, ..., σ_n} be the set of embeddings of K into C which are the identity on L. If b₁, ..., b_n is any basis of K over L, let d(b₁, ..., b_n) be the square of the determinant of the n by n matrix whose (i,j)-entry is σ_i(b_j). Then, the relative discriminant of K/L is the ideal generated by the d(b₁, ..., b_n) as {b₁, ..., b_n} varies over all integral bases of K/L. (i.e. bases with the property that b_i ∈ O_K 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 Δ_K/Q is the principal ideal of Z generated by the absolute discriminant Δ_K . 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 Δ_K/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 number field, K, of degree n, often denoted rd_K, is defined as the n-th root of the absolute value of the (absolute) discriminant of K. The relation between relative discriminants in a tower of fields shows that the root discriminant does not change in an unramified extension. The existence of a class field tower provides bounds on the root discriminant: the existence of an infinite class field tower over Q(√-m) where m = 3·5·7·11·19 shows that there are infinitely many fields with root discriminant 2√m ≈ 296.276. If we let r and 2s be the number of real and complex embeddings, so that n = r + 2s, put ρ = r/n and σ = 2s/n. Set α(ρ, σ) to be the infimum of rd_K for K with (r', 2s') = (ρn, σn). We have (for all n large enough)

$\alpha(\rho,\sigma) \ge 60.8^\rho 22.3^\sigma$

and on the assumption of the generalized Riemann hypothesis

$\alpha(\rho,\sigma) \ge 215.3^\rho 44.7^\sigma .$

So we have α(0,1) < 296.276. Martinet has shown α(0,1) < 93 and α(1,0) < 1059. Voight 2008 proves that for totally real fields, the root discriminant is > 14, with 1229 exceptions.

Relation to other quantities

• When embedded into $K\otimes_\mathbf{Q}\mathbf{R}$, the volume of the fundamental domain of O_K is $\sqrt{|\Delta_K|}$ (sometimes a different measure is used and the volume obtained is $2^{-r_2}\sqrt{|\Delta_K|}$, 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.

References

Secondary sources

• . Translated from the original French by John Meldrum