# 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.

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*.

## Definition

Let *K* be an algebraic number field, and let *O*_{K} be its ring of integers. Let *b*_{1}, ..., *b*_{n} be an integral basis of *O*_{K} (i.e. a basis as a **Z**-module), and let {σ_{1}, ..., 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,

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*_{i}*b*_{j}). This matrix equals *B*^{T}*B*, 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 is[2]

- An integer that occurs as the discriminant of a quadratic number field is called a fundamental discriminant.[3]

- Cyclotomic fields: let
*n*> 2 be an integer, let ζ_{n}be a primitive*n*th root of unity, and let*K*_{n}=**Q**(ζ_{n}) be the*n*th cyclotomic field. The discriminant of*K*_{n}is given by[2][4]

- where 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 choose the integral basis of*O*_{K}to be*b*_{1}= 1,*b*_{2}= α,*b*_{3}=*α*^{2}, ...,*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

- 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*x*^{3}−*x*^{2}− 2*x*− 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.[5][6] - 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
*x*^{3}− 21*x*+ 28 or*x*^{3}− 21*x*− 35, respectively.[7]

## Basic results

**Brill's theorem**:[8] The sign of the discriminant is (−1)^{r2}where*r*_{2}is the number of complex places of*K*.[9]- A prime
*p*ramifies in*K*if and only if*p*divides Δ_{K}.[10] **Stickelberger's theorem**:[11]

**Minkowski's bound**:[12] Let*n*denote the degree of the extension*K*/**Q**and*r*_{2}the number of complex places of*K*, then

**Minkowski's theorem**:[13] If*K*is not**Q**, then |Δ_{K}| > 1 (this follows directly from the Minkowski bound).**Hermite–Minkowski theorem**:[14] 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 with Hermite's theorem (that there are only finitely many algebraic number fields with prescribed discriminant).

## History

The definition of the discriminant of a general algebraic number field, *K*, was given by Dedekind in 1871.[15] At this point, he already knew the relationship between the discriminant and ramification.[16]

Hermite's theorem predates the general definition of the discriminant with Charles Hermite publishing a proof of it in 1857.[17] In 1877, Alexander von Brill determined the sign of the discriminant.[18] Leopold Kronecker first stated Minkowski's theorem in 1882,[19] though the first proof was given by Hermann Minkowski in 1891.[20] In the same year, Minkowski published his bound on the discriminant.[21] Near the end of the nineteenth century, Ludwig Stickelberger obtained his theorem on the residue of the discriminant modulo four.[22][23]

## 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 {σ_{1}, ..., σ_{n}} be the set of embeddings of *K* into **C** which are the identity on *L*. If *b*_{1}, ..., *b*_{n} is any basis of *K* over *L*, let *d*(*b*_{1}, ..., *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*(*b*_{1}, ..., *b*_{n}) as {*b*_{1}, ..., *b*_{n}} varies over all integral bases of *K*/*L*. (i.e. bases with the property that *b _{i}* ∈

*O*for all

_{K}*i*.) Alternatively, the relative discriminant of

*K*/

*L*is the norm of the different of

*K*/

*L*.[24] 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

where denotes relative norm.[25]

### 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.[24] 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*.[26] 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.[27] If we let *r* and 2*s* be the number of real and complex embeddings, so that *n* = *r* + 2*s*, put *ρ* = *r*/*n* and *σ* = 2*s*/*n*. Set *α*(*ρ*, *σ*) to be the infimum of rd_{K} for *K* with (*r', 2*s') = (*ρn*, *σn*). We have (for all n large enough) [27]

and on the assumption of the generalized Riemann hypothesis

So we have *α*(0,1) < 296.276. Martinet has shown *α*(0,1) < 93 and *α*(1,0) < 1059.[27][28] Voight 2008 proves that for totally real fields, the root discriminant is > 14, with 1229 exceptions.

## Relation to other quantities

- When embedded into , the volume of the fundamental domain of
*O*_{K}is (sometimes a different measure is used and the volume obtained is , where*r*_{2}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.[29]

## Notes

- Cohen, Diaz y Diaz & Olivier 2002
- Manin, Yu. I.; Panchishkin, A. A. (2007),
*Introduction to Modern Number Theory*, Encyclopaedia of Mathematical Sciences,**49**(Second ed.), p. 130, ISBN 978-3-540-20364-3, ISSN 0938-0396, Zbl 1079.11002 - Definition 5.1.2 of Cohen 1993
- Proposition 2.7 of Washington 1997
- Dedekind 1878, pp. 30–31
- Narkiewicz 2004, p. 64
- Cohen 1993, Theorem 6.4.6
- Koch 1997, p. 11
- Lemma 2.2 of Washington 1997
- Corollary III.2.12 of Neukirch 1999
- Exercise I.2.7 of Neukirch 1999
- Proposition III.2.14 of Neukirch 1999
- Theorem III.2.17 of Neukirch 1999
- Theorem III.2.16 of Neukirch 1999
- Dedekind's supplement X of the second edition of Peter Gustav Lejeune Dirichlet's Vorlesungen über Zahlentheorie (Dedekind 1871)
- Bourbaki 1994
- Hermite 1857.
- Brill 1877.
- Kronecker 1882.
- Minkowski 1891a.
- Minkowski 1891b.
- Stickelberger 1897.
- All facts in this paragraph can be found in Narkiewicz 2004, pp. 59, 81
- Neukirch 1999, §III.2
- Corollary III.2.10 of Neukirch 1999 or Proposition III.2.15 of Fröhlich & Taylor 1993
- Voight 2008
- Koch 1997, pp. 181–182
- Martinet, Jacques (1978). "Tours de corps de classes et estimations de discriminants".
*Inventiones Mathematicae*(in French).**44**: 65–73. Bibcode:1978InMat..44...65M. doi:10.1007/bf01389902. Zbl 0369.12007. - Section 4.4 of Serre 1967

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## Further reading

- Milne, James S. (1998),
*Algebraic Number Theory*, retrieved 2008-08-20