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Discriminant of an algebraic number field

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 squared volume of the fundamental domain of the ring of integers, and it regulates which primes are ramified.

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

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. A 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.[1]


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.

: let d be a square-free integer, then the discriminant of is[2]

Quadratic number fields

Brill's theorem: The sign of the discriminant is (−1)r2 where r2 is the number of complex places of K.[9]

[8]

A prime p ramifies in K if and only if p divides ΔK .[11]

[10]

Stickelberger's theorem:

[12]

When embedded into , the volume of the fundamental domain of OK is (sometimes a different is used and the volume obtained is , where r2 is the number of complex places of K).

measure

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 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.[31]

Artin conductor

Brill, Alexander von (1877), , Mathematische Annalen, 12 (1): 87–89, doi:10.1007/BF01442468, JFM 09.0059.02, MR 1509928, S2CID 120947279, retrieved 2009-08-22

"Ueber die Discriminante"

(1871), Vorlesungen über Zahlentheorie von P.G. Lejeune Dirichlet (2 ed.), Vieweg, retrieved 2009-08-05

Dedekind, Richard

(1878), "Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Congruenzen", Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 23 (1), retrieved 2009-08-20

Dedekind, Richard

(1882), "Grundzüge einer arithmetischen Theorie der algebraischen Grössen", Crelle's Journal, 92: 1–122, JFM 14.0038.02, retrieved 2009-08-20

Kronecker, Leopold

(1891a), "Ueber die positiven quadratischen Formen und über kettenbruchähnliche Algorithmen", Crelle's Journal, 1891 (107): 278–297, doi:10.1515/crll.1891.107.278, JFM 23.0212.01, retrieved 2009-08-20

Minkowski, Hermann

(1891b), "Théorèmes d'arithmétiques", Comptes rendus de l'Académie des sciences, 112: 209–212, JFM 23.0214.01

Minkowski, Hermann

(1897), "Über eine neue Eigenschaft der Diskriminanten algebraischer Zahlkörper", Proceedings of the First International Congress of Mathematicians, Zürich, pp. 182–193, JFM 29.0172.03

Stickelberger, Ludwig

(1998), Algebraic Number Theory, retrieved 2008-08-20

Milne, James S.