Discriminant of an algebraic number field#Basic results

Let $K$ be an algebraic number field, and let $\backslash mathcal\{O\}\_K$ be its ring of integers. Let $b\_1,\backslash dots,b\_n$ be an integral basis of $\backslash mathcal\{O\}\_K$ (i.e. a basis as a $\backslash Z$-module), and let $\backslash \{\backslash sigma\_1,\backslash dots,\backslash sigma\_n\backslash \}$ be the set of embeddings of $K$ into the complex numbers (i.e. injective ring homomorphisms $K\backslash to\backslash C$). The discriminant of $K$ is the square of the determinant of the $n\backslash times\; n$ matrix $B$ whose $(i,j)$-entry is $\backslash sigma\_i(b\_j)$. Symbolically,

:$\backslash Delta\_K=\backslash det\backslash left(\backslash begin\{array\}\{cccc\}$

\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 $\backslash Q$ can be used. Specifically, define the trace form to be the matrix whose $(i,j)$-entry is $\backslash operatorname\{Tr\}\_\{K/\backslash Q\}(b\_ib\_j)$. This matrix equals $B^T\backslash !B$, so the square of the discriminant of $K$ is the determinant of this matrix.

The discriminant of an order in $K$ with integral basis $b\_1,\backslash dots,b\_n$ is defined in the same way.

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

::

:An integer that occurs as the discriminant of a quadratic number field is called a fundamental discriminant.Definition 5.1.2 of {{harvnb|Cohen|1993}}

• Cyclotomic fields: let $n>2$ be an integer, let $\backslash zeta\_n$ be a primitive n-th root of unity, and let $K\_n=\backslash Q(\backslash zeta\_n)$ be the $n$-th cyclotomic field. The discriminant of $K\_n$ is given by{{citation | first1=Yu. I. | last1=Manin | author-link1=Yuri I. Manin | first2=A. A. | last2=Panchishkin | title=Introduction to Modern Number Theory | series=Encyclopaedia of Mathematical Sciences | volume=49 | edition=Second | year=2007 | isbn=978-3-540-20364-3 | issn=0938-0396 | zbl=1079.11002 | page=130 }}Proposition 2.7 of {{harvnb|Washington|1997}}

:: $\backslash Delta\_\{K\_n\}\; =\; (-1)^\{\backslash varphi(n)/2\}\; \backslash frac\{n^\{\backslash varphi(n)\}\}\{\backslash displaystyle\backslash prod\_\{p|n\}\; p^\{\backslash varphi(n)/(p-1)\}\}$

: where $\backslash varphi(n)$ 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 $\backslash mathcal\{O\}\_K$ of algebraic integers has a power integral basis, that is, can be written as $\backslash mathcal\{O\}\_K=\backslash Z[\backslash alpha]$, the discriminant of $K$ is equal to the discriminant of the minimal polynomial of $\backslash alpha$. To see this, one can choose the integral basis of $\backslash mathcal\{O\}\_K$ to be

::$b\_1=1,\; b\_2=\backslash alpha,\; b\_3=\backslash alpha^2,\; \backslash dots,b\_n\; =\; \backslash alpha^\{n-1\}$.

:Then, the matrix $B$ in the definition is the Vandermonde matrix associated to $\backslash alpha\_i=\backslash sigma\_i(\backslash alpha)$, whose squared determinant is

::

:which is exactly the definition of the discriminant of the minimal polynomial.

• Let $K=\backslash Q(\backslash alpha)$ be the number field obtained by adjoining a root $\backslash alpha$ of the polynomial $x^3-x^2-2x-8$. 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

::$\backslash left\backslash \{1,\backslash alpha,\backslash frac\{\backslash alpha(\backslash alpha+1)\}\{2\}\backslash right\backslash \}$

:and the discriminant of $K$ is −503.{{harvnb|Dedekind|1878}}, pp. 30–31{{harvnb|Narkiewicz|2004}}, p. 64

• 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 $x^3-21x+28$ or $x^3-21x-35$, respectively.{{harvnb|Cohen|1993|loc=Theorem 6.4.6}}

• Brill's theorem:{{harvnb|Koch|1997|p=11}} The sign of the discriminant is $(-1)^\{r\_2\}$ where $r\_2$ is the number of complex places of $K$.Lemma 2.2 of {{harvnb|Washington|1997}}
• A prime $p$ ramifies in $K$ if and only if $p$ divides $\backslash Delta\_K$.Corollary III.2.12 of {{harvnb|Neukirch|1999}}{{cite web|title=Discriminants and ramified primes|last=Conrad|first=Keith|url=https://kconrad.math.uconn.edu/blurbs/gradnumthy/disc.pdf#theorem.1.3|quote=Theorem 1.3 (Dedekind). For a number field K, a prime p ramifies in K if and only if p divides the integer discZ(OK)}}
• Stickelberger's theorem:Exercise I.2.7 of {{harvnb|Neukirch|1999}}

:: $\backslash Delta\_K\backslash equiv\; 0\backslash text\{\; or\; \}1\; \backslash pmod\; 4.$

• Minkowski's bound:Proposition III.2.14 of {{harvnb|Neukirch|1999}} Let $n$ denote the degree of the extension $K/\backslash Q$ and $r\_2$ the number of complex places of $K$, then

:: $|\backslash Delta\_K|^\{1/2\}\backslash geq\; \backslash frac\{n^n\}\{n!\}\backslash left(\backslash frac\{\backslash pi\}\{4\}\backslash right)^\{r\_2\}\; \backslash geq\; \backslash frac\{n^n\}\{n!\}\backslash left(\backslash frac\{\backslash pi\}\{4\}\backslash right)^\{n/2\}.$

• Minkowski's theorem:Theorem III.2.17 of {{harvnb|Neukirch|1999}} If $K$ is not $\backslash Q$, then $|\backslash Delta\_K|>1$ (this follows directly from the Minkowski bound).
• Hermite–Minkowski theorem:Theorem III.2.16 of {{harvnb|Neukirch|1999}} Let $N$ be a positive integer. There are only finitely many (up to isomorphisms) algebraic number fields $K$ with

File:Dedekind.jpeg showed that every number field possesses an integral basis, allowing him to define the discriminant of an arbitrary number field.Dedekind's supplement X of the second edition of Peter Gustav Lejeune Dirichlet's Vorlesungen über Zahlentheorie {{harv|Dedekind|1871}}]]

The definition of the discriminant of a general algebraic number field, K, was given by Dedekind in 1871. At this point, he already knew the relationship between the discriminant and ramification.{{harvnb|Bourbaki|1994}}

Hermite's theorem predates the general definition of the discriminant with Charles Hermite publishing a proof of it in 1857.{{sfn|Hermite|1857}} In 1877, Alexander von Brill determined the sign of the discriminant.{{sfn|Brill|1877}} Leopold Kronecker first stated Minkowski's theorem in 1882,{{sfn|Kronecker|1882}} though the first proof was given by Hermann Minkowski in 1891.{{sfn|Minkowski|1891a}} In the same year, Minkowski published his bound on the discriminant.{{sfn|Minkowski| 1891b}} Near the end of the nineteenth century, Ludwig Stickelberger obtained his theorem on the residue of the discriminant modulo four.{{sfn|Stickelberger|1897}}All facts in this paragraph can be found in {{harvnb|Narkiewicz|2004|pp=59, 81}}

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.{{harvnb|Neukirch|1999|loc=§III.2}} 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

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

where $\backslash mathcal\{N\}$ denotes relative norm.Corollary III.2.10 of {{harvnb|Neukirch|1999}} or Proposition III.2.15 of {{harvnb|Fröhlich|Taylor|1993}}

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. 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

:$\backslash operatorname\{rd\}\_K\; =\; |\backslash Delta\_K|^\{1/n\}.${{cite journal| last1=Hajir | first1=Farshid | last2=Maire | first2=Christian | title=Tamely ramified towers and discriminant bounds for number fields. II | journal=J. Symbolic Comput. | volume=33 | pages=415–423 | year=2002 | doi=10.1023/A:1017537415688 | doi-access=free }}

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

:$\backslash alpha(\backslash rho,\backslash sigma)\; \backslash ge\; 60.8^\backslash rho\; 22.3^\backslash sigma$,

and the generalized Riemann hypothesis implies the stronger bound

:$\backslash alpha(\backslash rho,\backslash sigma)\; \backslash ge\; 215.3^\backslash rho\; 44.7^\backslash sigma\; .${{harvnb|Koch|1997|pp=181–182}}

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.{{harvnb|Voight|2008}}

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({{radic|-m}}) with m = 3·5·7·11·19 produces fields of arbitrarily large degree with root discriminant 2{{radic|m}} ≈ 296.276, 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, improving upon earlier bounds of Martinet.{{cite journal | zbl=0369.12007 | last=Martinet | first=Jacques | title=Tours de corps de classes et estimations de discriminants | language=fr | journal=Inventiones Mathematicae | volume=44 | pages=65–73 | year=1978 | doi=10.1007/bf01389902| bibcode=1978InMat..44...65M | s2cid=122278145 }}

• When embedded into $K\backslash otimes\_\backslash mathbf\{Q\}\backslash mathbf\{R\}$, the volume of the fundamental domain of O_K is $\backslash sqrt$

  \Delta_K

  (sometimes a different measure is used and the volume obtained is $2^\{-r\_2\}\backslash sqrt$

  \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.Section 4.4 of {{harvnb|Serre|1967}}

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Category:Algebraic number theory