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]

1 Definition
2 Examples
3 Important Results
4 Relative Discriminant
5 Relation to Other Quantities
6 References
7 Citations
8 Further Reading

[edit] 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. 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 σ_i(b_j). 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 Tr_K/Q(b_ib_j). Then the discriminant of K is the determinant of this matrix.

[edit] Examples

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

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

Cyclotomic fields: let n be a positive 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^[2]

$\Delta_{K_n}=(-1)^{\phi(n)/2}\frac{n^{\phi(n)}}{\prod_{p|n}p^{\phi(n)/(p-1)}}$
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 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 = α^n-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 − 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 Δ_K.^[4]
Stickelberger's Theorem:^[5]

$\Delta_K\equiv 0\mbox{ or }1 \mbox{ mod 4}$

Minkowski bound:^[6] Let n denote the degree of the extension K/Q, then

$|\Delta_K|^{1/2}\geq \frac{n^n}{n!}\left(\frac{\pi}{4}\right)^{n/2}$ .

Minkowski's Theorem:^[7] If K is not Q, then |Δ_K| > 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 Δ_K < 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 Δ_K/L of an extension of number fields K/L, which is an ideal in O_L. When L = Q, the relative discriminant Δ_K/Q is the principal ideal generated by the absolute discriminant Δ_K.

[edit] 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 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] References

^ Cohen et al. 2002
^ Proposition 2.7 of Washington 1997
^ Lemma 2.2 of Washington 1997
^ Corollary 2.12 of Neukirch 1999
^ Exercise 1.2.7 of Neukirch 1999
^ Proposition 2.14 of Neukirch 1999
^ Theorem 2.17 of Neukirch 1999
^ Theorem 2.16 of Neukirch 1999

[edit] Citations

Cohen, Henri; Diaz y Diaz, Francisco & Olivier, Michel (2002), “A Survey of Discriminant Counting”, Algorithmic Number Theory, Fifth International Syposium, vol. 2369, Lecture Notes in Computer Science, Berlin, New York: Springer-Verlag, pp. 80-94, MR 0302-9743, ISSN 0302-9743
Neukirch, Jürgen (1999), Algebraic Number Theory, vol. 322, Grundlehren der mathematschen Wissenschaften, Berlin, New York: Springer-Verlag, MR 1697859, ISBN 3-540-65399-6
Washington, Lawrence (1997), Introduction to Cyclotomic Fields, vol. 83, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, MR 1421575, ISBN 0-387-94762-0