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Discriminant of an algebraic number field

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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 related 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's 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.[1]

Contents

[edit] Definition

Let K be an algebraic number field, and let OK be its ring of integers. Let b1, ..., bn be an integral basis of OK (i.e. a basis as a Z-module), and let {σ1, ..., σn} be the set of embeddings of K into the complex numbers (i.e. 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 σi(bj). Symbolically,

\Delta_K=\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)^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 TrK/Q(bibj): we may identify this as BTB. Then the discriminant of K is the determinant of this matrix.

[edit] Examples

\Delta_K=\left\{\begin{array}{ll} d &\mbox{if }d\equiv 1\mbox{ mod }4 \\ 4d &\mbox{if }d\equiv 2,3\mbox{ mod }4. \\\end{array}\right.
An integer that occurs as the discriminant of a quadratic number field is called a fundamental discriminant.[2]
\Delta_{K_n} = (-1)^{\varphi(n)/2} \frac{n^{\varphi(n)}}{\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 OK = 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 OK to be b1 = 1, b2 = α, b3 = α2, ..., bn = αn−1. Then, the matrix in the definition is the Vandermonde matrix associated to αi = σi(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 − 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.

[edit] Important results

  • The sign of the discriminant is (−1)r2 where r2 is the number of complex places of K.[4]
  • A prime p ramifies in K if, and only if, p divides ΔK.[5]
  • Stickelberger's Theorem:[6]
\Delta_K\equiv 0\mbox{ or }1 \mbox{ mod 4}
  • Minkowski bound:[7] 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:[8] If K is not Q, then |ΔK| > 1 (this follows directly from the Minkowski bound).
  • Hermite's Theorem:[9] Let N be a positive integer. There are only finitely many algebraic number fields K with ΔK < N.

[edit] 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 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[10].

[edit] Ramification

The relative discriminant encodes 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.

[edit] Root discriminant

The root discriminant of K, often denoted rdK, is defined as the n-th root of the absolute value of the (absolute) discriminant of K[11].

[edit] Relation to other quantities

  • When embedded into K\otimes_\mathbf{Q}\mathbf{R}, the volume of the fundamental domain of OK 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 is related to the Artin conductors of the characters of the Galois group of K/L through the conductor-discriminant formula.

[edit] Notes

  1. ^ Cohen et al. 2002
  2. ^ Definition 5.1.2 of Cohen 1993
  3. ^ Proposition 2.7 of Washington 1997
  4. ^ Lemma 2.2 of Washington 1997
  5. ^ Corollary III.2.12 of Neukirch 1999
  6. ^ Exercise I.2.7 of Neukirch 1999
  7. ^ Proposition III.2.14 of Neukirch 1999
  8. ^ Theorem III.2.17 of Neukirch 1999
  9. ^ Theorem III.2.16 of Neukirch 1999
  10. ^ Corollary III.2.10 of Neukirch 1999 or Proposition III.2.15 of Frohlich and Taylor 1991
  11. ^ Voight 2008

[edit] References

[edit] Further reading

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