Number field. (LMFDB)

A number field is a field extension of finite degree of the field \Q of rational numbers. In the LMFDB, number fields are identified by a label.

Depends on: Definition 2.12

Abelian number field. (LMFDB)

A number field K is abelian if it is Galois over \Q and its Galois group \Gal(K/\Q) is abelian.

Depends on: Definition 2.1 Definition 2.21

Absolute discriminant of a number field. (LMFDB)

The absolute discriminant of a number field is the absolute value of its discriminant.

Depends on: Definition 2.1 Definition 2.15

Absolute value of a field. (LMFDB)

An absolute value of a field k is a function |\ |:k\to \R_{\ge 0} that satisfies:

• |x|=0 if and only if x=0;

• |xy| = |x||y|;

• |x+y| \le |x|+|y|.

Absolute values that satisfy the stronger condition |x+y|\le \max(|x|,|y|) are nonarchimedean, while those that do not are archimedean; the latter arise only in fields of characteristic zero. The trivial absolute value assigns 1 to every nonzero element of k; it is a nonarchimedean absolute value.

Absolute values |\ |_1 and |\ |_2 are equivalent if there exists a positive real number c such that |x|_1 = |x|_2^c for all x\in k; this defines an equivalence relation on the set of absolute values of k.

Arithmetically equivalent fields. (LMFDB)

Two number fields are arithmetically equivalent if they have the same Dedekind \zeta-functions. Arithmetically equivalent fields share many invariants, such as their degrees, signatures, discriminants, and Galois groups. For a given field, the existence of an arithmetically equivalent sibling depends only on the Galois group.

Depends on: Definition 2.1 Definition 2.12 Definition 2.15 Definition 2.21 Definition 2.61 Definition 2.62

Class number of a number field. (LMFDB)

The class number of a number field K is the order of the ideal class group of K.

Depends on: Definition 2.1 Definition 2.24

Analytic class number formula. (LMFDB)

If K is a number field with signature (r_1, r_2), discriminant D, regulator R, class number h, containing w roots of unity, and Dedekind \zeta-function \zeta_K, then \zeta_K has a meromorphic continuation to the whole complex plane with a single pole at s=1, which is of order 1. The analytic class number formula gives the residue at this pole:

\lim_{s\to 1}\ (s-1)\zeta_K(s) = \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}} .

Depends on: Definition 2.1 Definition 2.6 Definition 2.15 Definition 2.54 Definition 2.62

CM number field. (LMFDB)

A CM field is a totally complex quadratic extension of a totally real number field.

Depends on: Definition 2.1 Definition 2.65 Definition 2.67

Complex embedding. (LMFDB)

A complex embedding of a number field K is a nonzero field homomorphism K\to \C whose image is not contained in \R.

A single number field may have several distinct complex embeddings.

For K=\Q(a) where a is an algebraic number with minimal polynomial f(X), the embeddings \iota:K\to\C are determined by the value z=\iota(a) which is one of the complex roots of f(X), and the embedding is complex when z\notin\R. The complex embeddings come in conjugate pairs.

Depends on: Definition 2.1 Definition 2.35

Conductor of an abelian number field. (LMFDB)

If a number field K is abelian, then K\subseteq \Q(\zeta_n) for some positive integer n. The minimum such n is the conductor of K.

Depends on: Definition 2.1 Definition 2.2

Defining Polynomial of a Number Field. (LMFDB)

A defining polynomial of a number field K is an irreducible polynomial f\in\Q[x] such that K\cong \mathbb{Q}(a), where a is a root of f(x). Equivalently, it is a polynomial f\in \Q[x] such that K \cong \Q[x]/(f).

A root a \in K of the defining polynomial is a generator of K.

Depends on: Definition 2.1

Degree of a number field. (LMFDB)

The degree of a number field K is its degree as an extension of the rational field \mathbb{Q}, i.e., the dimension of K as a \mathbb{Q}-vector space. The degree of K/\Q is written [K:\mathbb{Q}].

Depends on: Definition 2.13

Number field degree (mathlib). (LMFDB)

This is also defined in mathlib as 'finrank ℚ K', where K is the number field and 'finrank ℚ' is its module rank over the rationals.

Dirichlet group of an Abelian number field. (LMFDB)

If K is an abelian number field, then K\subseteq \Q(\zeta_n) for some positive integer n. Take the minimal such n, i.e., the conductor of K.

The Galois group \Gal(\Q(\zeta_n)/\Q) is canonically isomorphic to \Z_n^\times. The Dirichlet characters modulo n form the dual group of homomorphisms \chi:\Z_n^\times\to\C^\times. Since \Gal(K/\Q) is a quotient group of \Gal(\Q(\zeta_n)/\Q), its dual group is a subgroup of the group of Dirichlet characters modulo n, called the Dirichlet character group of K.

Depends on: Definition 1.45 Definition 2.2 Definition 2.10

Discriminant of a number field. (LMFDB)

The discriminant of a number field K is the square of the determinant of the matrix

\left( \begin{array}{ccc} \sigma_1(\beta_1) & \cdots & \sigma_1(\beta_n) \\ \vdots & & \vdots \\ \sigma_n(\beta_1) & \cdots & \sigma_n(\beta_n) \\ \end{array} \right)

where \sigma_1,..., \sigma_n are the embeddings of K into the complex numbers \mathbb{C}, and \{\beta_1, \ldots, \beta_n\} is an integral basis for the ring of integers of K.

The discriminant of K is a non-zero integer divisible exactly by the primes which ramify in K.

Depends on: Definition 2.1 Definition 2.29

Discriminant root field. (LMFDB)

If K/F is a finite algebraic extension, it can be defined by a polynomial f(x)\in F[x]. The polynomial discriminant, \mathrm{disc}(f), is well-defined up a factor of a non-zero square. The discriminant root field of the extension is F(\sqrt{\mathrm{disc}(f)}), which is well-defined.

If n=[K:F], then the Galois group G for K/F is a subgroup of S_n, well-defined up to conjugation. The discriminant root field can alternatively be described as the fixed field of G\cap A_n.

Embedding of a number field. (LMFDB)

An embedding of a number field K is a field homomorphism K\to \C. A number field of degree n has n distinct embeddings, which may be distinguished as real or complex depending on whether the image of the embedding is contained in \R or not.

Complex embeddings necessarily come in conjugate pairs. The signature of a number field is determined by the number of real embeddings and the number of pairs of conjugate complex embeddings.

For K=\Q(a), where a is an algebraic number with minimal polynomial f(X), each embedding \iota is uniquely determined by the value z=\iota(a), which is one of the complex roots of f(X). The embedding is real if z\in\R and complex if z\notin\R.

Depends on: Definition 2.1 Definition 2.9 Definition 2.12 Definition 2.35 Definition 2.51 Definition 2.62

Frobenius cycle types. (LMFDB)

If K is a degree n extension of \mathbb{Q}, \hat{K} its normal closure and G=\text{Gal}(\hat{K}/\mathbb{Q}), then G acts on the set of n embeddings of K\to \hat{K} giving an embedding G\to S_n. Let \mathcal{O}_K be the ring of integers of K and p a prime number. Then

p\mathcal{O}_K = P_1^{e_1}\cdots P_g^{e_g}

where the P_i are distinct prime ideals of \mathcal{O}_K. The prime p is unramified if e_i=1 for all i.

Suppose hereafter that p is unramified. For each P_i, there is a unique element of G that fixes P_i and acts on the quotient \mathcal{O}_K/P_i via the Frobenius automorphism x \mapsto x^p; this element is the Frobenius element associated to P_i. The Frobenius elements associated to the different P_i are conjugate to each other, so their images in S_n all have the same lengths of cycles in their disjoint cycle decompositions. This is the Frobenius cycle type of p.

Alternatively, for each prime P_i, its residue degree f_i is defined by |\mathcal{O}_K/P_i| = p^{f_i}. The list of f_i is the same partition of n as the cycle decomposition described above.

Fundamental units of a number field. (LMFDB)

A minimal set of generators of a maximal torsion-free subgroup of the unit group of a number field K is called a set of fundamental units for K.

Depends on: Definition 2.1 Definition 2.68

Galois closure of an extension. (LMFDB)

If K is a separable algebraic extension of a field F, then its Galois closure is the smallest extension field, in terms of inclusion, which contains K and is Galois over F. If K=F(\alpha) where \alpha has irreducible polynomial f over F, then the Galois closure of K is the splitting field of f over F.

Depends on: Definition 2.58