Discriminant of an algebraic number field

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. 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,

${\displaystyle \Delta _{K}=\det \left({\begin{array}{cccc}\sigma _{1}(b_{1})&\sigma _{1}(b_{2})&\cdots &\sigma _{1}(b_{n})\\\sigma _{2}(b_{1})&\ddots &&\vdots \\\vdots &&\ddots &\vdots \\\sigma _{n}(b_{1})&\cdots &\cdots &\sigma _{n}(b_{n})\end{array}}\right)^{2}.}$ 

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). This matrix equals B^TB, so the square of the discriminant of K is the determinant of this matrix.

The discriminant of an order in K with integral basis b₁, ..., b_n is defined in the same way.

• Quadratic number fields: let d be a square-free integer, then the discriminant of ${\displaystyle K=\mathbf {Q} ({\sqrt {d}})}$  is^[2]

${\displaystyle \Delta _{K}=\left\{{\begin{array}{ll}d&{\text{if }}d\equiv 1{\pmod {4}}\\4d&{\text{if }}d\equiv 2,3{\pmod {4}}.\\\end{array}}\right.}$ 

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 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][4]

${\displaystyle \Delta _{K_{n}}=(-1)^{\varphi (n)/2}{\frac {n^{\varphi (n)}}{\displaystyle \prod _{p|n}p^{\varphi (n)/(p-1)}}}}$ 

where ${\displaystyle \varphi (n)}$  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 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 b₁ = 1, b₂ = α, b₃ = α², ..., b_n = αⁿ⁻¹. Then, the matrix in the definition is the Vandermonde matrix associated to α_i = σ_i(α), whose determinant squared is

${\displaystyle \prod _{1\leq i<j\leq n}(\alpha _{i}-\alpha _{j})^{2}}$ 

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 − x² − 2x − 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 x³ − 21x + 28 or x³ − 21x − 35, respectively.^[7]

• Brill's theorem:^[8] The sign of the discriminant is (−1)^r₂ where r₂ is the number of complex places of K.^[9]
• A prime p ramifies in K if and only if p divides Δ_K .^[10][11]
• Stickelberger's theorem:^[12]

${\displaystyle \Delta _{K}\equiv 0{\text{ or }}1{\pmod {4}}.}$ 

• Minkowski's bound:^[13] Let n denote the degree of the extension K/Q and r₂ the number of complex places of K, then

${\displaystyle |\Delta _{K}|^{1/2}\geq {\frac {n^{n}}{n!}}\left({\frac {\pi }{4}}\right)^{r_{2}}\geq {\frac {n^{n}}{n!}}\left({\frac {\pi }{4}}\right)^{n/2}.}$ 

• Minkowski's theorem:^[14] If K is not Q, then |Δ_K| > 1 (this follows directly from the Minkowski bound).
• Hermite–Minkowski theorem:^[15] 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).

The definition of the discriminant of a general algebraic number field, K, was given by Dedekind in 1871.^[16] At this point, he already knew the relationship between the discriminant and ramification.^[17]

Hermite's theorem predates the general definition of the discriminant with Charles Hermite publishing a proof of it in 1857.^[18] In 1877, Alexander von Brill determined the sign of the discriminant.^[19] Leopold Kronecker first stated Minkowski's theorem in 1882,^[20] though the first proof was given by Hermann Minkowski in 1891.^[21] In the same year, Minkowski published his bound on the discriminant.^[22] Near the end of the nineteenth century, Ludwig Stickelberger obtained his theorem on the residue of the discriminant modulo four.^[23][24]

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 {σ₁, ..., σ_n} be the set of embeddings of K into C which are the identity on L. If b₁, ..., b_n is any basis of K over L, let d(b₁, ..., 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(b₁, ..., b_n) as {b₁, ..., b_n} varies over all integral bases of K/L. (i.e. bases with the property that b_i ∈ O_K for all i.) Alternatively, the relative discriminant of K/L is the norm of the different of K/L.^[25] 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

${\displaystyle \Delta _{K/F}={\mathcal {N}}_{L/F}\left({\Delta _{K/L}}\right)\Delta _{L/F}^{[K:L]}}$ 

where ${\displaystyle {\mathcal {N}}}$  denotes relative norm.^[26]

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.^[25] 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.

The root discriminant of a degree n number field K is defined by the formula

${\displaystyle \operatorname {rd} _{K}=|\Delta _{K}|^{1/n}.}$ ^[27]

The relation between relative discriminants in a tower of fields shows that the root discriminant does not change in an unramified extension.

Given nonnegative rational numbers ρ and σ, not both 0, and a positive integer n such that the pair (r,2s) = (ρn,σn) is in Z × 2Z, let α_n(ρ, σ) be the infimum of rd_K as K ranges over degree n number fields with r real embeddings and 2s complex embeddings, and let α(ρ, σ) =  liminf_n→∞ α_n(ρ, σ). Then

${\displaystyle \alpha (\rho ,\sigma )\geq 60.8^{\rho }22.3^{\sigma }}$ ,

and the generalized Riemann hypothesis implies the stronger bound

${\displaystyle \alpha (\rho ,\sigma )\geq 215.3^{\rho }44.7^{\sigma }.}$ ^[28]

There is also a lower bound that holds in all degrees, not just asymptotically: For totally real fields, the root discriminant is > 14, with 1229 exceptions.^[29]

On the other hand, the existence of an infinite class field tower can give upper bounds on the values of α(ρ, σ). For example, the infinite class field tower over Q(√-m) with m = 3·5·7·11·19 produces fields of arbitrarily large degree with root discriminant 2√m ≈ 296.276,^[28] so α(0,1) < 296.276. Using tamely ramified towers, Hajir and Maire have shown that α(1,0) < 954.3 and α(0,1) < 82.2,^[27] improving upon earlier bounds of Martinet.^[28][30]

• When embedded into ${\displaystyle K\otimes _{\mathbf {Q} }\mathbf {R} }$ , the volume of the fundamental domain of O_K is ${\displaystyle {\sqrt {|\Delta _{K}|}}}$  (sometimes a different measure is used and the volume obtained is ${\displaystyle 2^{-r_{2}}{\sqrt {|\Delta _{K}|}}}$ , 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 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.^[31]

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