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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]

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[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 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). 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.
\Delta_{K_n}=(-1)^{\phi(n)/2}\frac{n^{\phi(n)}}{\prod_{p|n}p^{\phi(n)/(p-1)}}
where φ(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 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.[3]
  • A prime p ramifies in K if, and only if, p divides ΔK.[4]
  • Stickelberger's Theorem:[5]
\Delta_K\equiv 0\mbox{ or }1 \mbox{ mod 4}
|\Delta_K|^{1/2}\geq \frac{n^n}{n!}\left(\frac{\pi}{4}\right)^{n/2}.
  • Mikowski's Theorem:[7] If K is not Q, then |ΔK| ≥ 1 (this follows directly from the Minkowski bound).
  • Hermite's Theorem:[8] 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. When L = Q, the relative discriminant ΔK/Q is the principal ideal generated by the absolute discriminant ΔK.

[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 of K/L is the norm of the different of K/L.
  • 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] Counting Discriminants

The number of quartic fields of bounded discriminant (blue crosses) and the asymptotic distribution (black line).
The number of quartic fields of bounded discriminant (blue crosses)[9] and the asymptotic distribution (black line).[10]

Hermite's theorem states that there are only finitely many algebraic number fields of bounded discriminant; the question of the exact number, for a given bound, has proved to be a difficult one. It has generally been attacked by fixing the degree of the number field (and also the Galois group of the Galois closure).

Similar conjectures exist for relative discriminants as well.

[edit] References

  1. ^ Cohen et al. 2002
  2. ^ Proposition 2.7 of Washington 1997
  3. ^ Lemma 2.2 of Washington 1997
  4. ^ Corollary 2.12 of Neukirch 1999
  5. ^ Exercise 1.2.7 of Neukirch 1999
  6. ^ Proposition 2.14 of Neukirch 1999
  7. ^ Theorem 2.17 of Neukirch 1999
  8. ^ Theorem 2.16 of Neukirch 1999
  9. ^ Data obtained in Cohen et al. 2003, and available at [1]
  10. ^ Pages 1031-1032 of Bhargava 2005

[edit] Citations

[edit] Further Reading