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

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

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

An absolute value of a field \(k\) is a function \(|\ |:k\to \mathbb {R}_{\ge 0}\) 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 \(k\).

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.

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

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:

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

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

A single number field may have several distinct complex embeddings.

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

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

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

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

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 \(K/\mathbb {Q}\) is written \([K:\mathbb {Q}]\).

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

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

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

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 \(K\).

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 \(G\cap A_n\).

An embedding of a number field \(K\) is a field homomorphism \(K\to \mathbb {C}\). 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 \(\mathbb {R}\) 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 \(K=\mathbb {Q}(a)\), 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 \(z\in \mathbb {R}\) and complex if \(z\notin \mathbb {R}\).

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

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.

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 \(K\).

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 \(F\).

Let \(K\) be a finite degree \(n\) separable extension of a field \(F\), and \(K^{gal}\) be its Galois (or normal) closure. The Galois group for \(K/F\) is the automorphism group \(\operatorname{Aut}(K^{gal}/F)\).

This automorphism group acts on the \(n\) embeddings \(K\hookrightarrow K^{gal}\) via composition. As a result, we get an injection \(\operatorname{Aut}(K^{gal}/F)\hookrightarrow S_n\), which is well-defined up to the labelling of the \(n\) embeddings, which corresponds to being well-defined up to conjugation in \(S_n\).

We use the notation \(\operatorname{Gal}(K/F)\) for \(\operatorname{Aut}(K/F)\) when \(K=K^{gal}\).

There is a naming convention for Galois groups up to degree \(47\).

The Galois root discriminant of a number field is the root discriminant of its Galois closure.

A generator of a number field \(K\) is an element \(a\in K\) such that \(K=\mathbb {Q}(a)\). The minimal polynomial of a generator is a defining polynomial for \(K\).

The ideal class group of a number field \(K\) with ring of integers \(O_K\) is the group of equivalence classes of ideals, given by the quotient of the multiplicative group of all fractional ideals of \(O_K\) by the subgroup of principal fractional ideals.

Since \(K\) is a number field, the ideal class group of \(K\) is a finite abelian group, and so has the structure of a product of cyclic groups encoded by a finite list \([a_1,\dots ,a_n]\), where the \(a_i\) are positive integers with \(a_i\mid a_{i+1}\) for \(1\leq i {\lt} n\).

In the LMFDB ideals in rings of integers of number fields are identified using the labeling system developed by John Cremona, Aurel Page and Andrew Sutherland [ .

In a number field \(K\), each nonzero ideal \(I\) of its ring of integers \(\mathcal{O}_K\) is assigned an ideal label of the form \(\texttt{N.i}\), where \(N\) and \(i\) are positive integers, in which \(N:=[\mathcal{O}_K:I]\) is the norm of the ideal and \(i\) is the index of the ideal in a sorted list of all ideals of norm \(N\). Once an integral primitive element \(\alpha \) for the field \(K\) is fixed, the ordering of ideals of the same norm is defined in a deterministic fashion (involving no arbitrary choices).

In the LMFDB we always represent number fields as \(K = \mathbb {Q}[X]/(g(X))\) where \(g\) is the unique monic integral polynomial which satisfies the polredabs condition. In this representation the image of \(X\) under the quotient map \(\mathbb {Q}[X]\rightarrow \mathbb {Q}[X]/(g(X))\) is a canonical integral primitive element \(\alpha \) for \(K\). Fixing this element determines a unique ordering of all \(\mathcal{O}_K\)-ideals of the same norm.

An inessential prime of a number field is a prime divisor of its index.

An element of a number field \(K\) is integral if it is integral over \(\mathbb {Z}\).

An integral basis of a number field \(K\) is a \(\mathbb {Z}\)-basis for the ring of integers of \(K\). This is also a \(\mathbb {Q}\)-basis for \(K\).

For a number field \(K\), intermediate fields \(F\) are fields with \(\mathbb {Q}\subsetneqq F \subsetneqq K\).

Let \(F\) be a subfield of \(K\),

and

Then \(K\) is Galois over \(F\) if \(K^{\operatorname{Aut}(K/F)} = F\).

Given a global number field \(K\) and a prime \(p\), the local algebra for \(K\) is \(K\otimes \mathbb {Q}_p\). This is a finite separable algebra over \(\mathbb {Q}_p\) which is isomorphic to a finite direct product of finite extension fields of \(\mathbb {Q}_p\).

The maximal CM subfield of a number field is the largest subfield by degree which is a CM field.

The minimal polynomial of an element \(a\) in a number field \(K\) is the unique monic polynomial \(f(X)\in \mathbb {Q}[X]\) of minimal degree such that \(f(a)=0\). It is necessarily irreducible over \(\mathbb {Q}.\)

The minimal sibling of a number field is a sibling that is minimal with respect to the following quantities considered in order:

• its degree

• the T-number of its Galois group

• the absolute value of its discriminant

• the vector \((a_0, a_1, \ldots , a_{n-1})\) of coefficients of its normalized defining polynomial

\(x^n+a_{n-1} x^{n-1}+\cdots +a_0\)

A number field \(K\) is monogenic if its ring of integers \(\mathcal{O}_K\) equals \(\mathbb {Z}[\alpha ]\) for some \(\alpha \in \mathcal{O}_K\).

A monomial order in a number field \(K\) is an order of the form \(\mathbb {Z}[\alpha ]\), where \(\alpha \) is an element of \(K\). The element \(\alpha \) is necessarily both an algebraic integer and a primitive element for \(K\).

The narrow class group (also called the strict class group) of a number field \(K\) is the group of equivalence classes of ideals, given by the quotient of the multiplicative group of all fractional ideals of \(K\) by the subgroup of principal fractional ideals which have a totally positive generator. It is a finite abelian group whose order is the narrow class number.

The narrow class number (also called the strict class number) of an algebraic number field is the order of its narrow class group. Since the ordinary ideal class group is a quotient of the narrow class group, the narrow class number is a multiple of the class number. Moreover, the ratio is a power of \(2\). The two class numbers are the same in many cases, for example when the number field is totally complex.

The LMFDB supports nicknames, short human-readable names for various fields. Examples include:

• Q, for the rationals \(\mathbb {Q}\)

• Qi, for \(\mathbb {Q}(i)\)

• QsqrtN, for \(\mathbb {Q}(\sqrt{N})\), as in Qsqrt-5 for \(\mathbb {Q}(\sqrt{-5})\)

• QzetaN, for \(\mathbb {Q}(\zeta _N)\), where \(\zeta _N\) is a primitive \(N\)th root of unity.

An order in a number field \(K\) is a subring of \(K\) which is also a lattice in \(K\). Every order in \(K\) is contained in the ring of integers of \(K\), which is itself an order in \(K\); for this reason, the ring of integers is sometimes called the maximal order.

Example: \(\mathbb {Z}[\sqrt{5}]\) is an order in \(K=\mathbb {Q}(\sqrt{5})\). However, it is not maximal, since the maximal order (i.e. ring of integers) of \(K\) is \(\mathbb {Z}\left[\frac{1+\sqrt{5}}2\right]\).

Let \(K\) be a number field, \(\mathcal{O}_K\) its ring of integers, \(\mathfrak {P}\) a non-zero prime ideal of \(\mathcal{O}_K\), and \(p\in \mathbb {Z}\cap \mathfrak {P}\). There are a couple of ways to construct \(K_{\mathfrak {P}}\), the \(p\)-adic completion of \(K\) at \(\mathfrak {P}\).

First, we can take the inverse limit

which is an integral domain. Its field of fractions is \(K_{\mathfrak {P}}\).

Second, since \(\mathcal{O}_K\) is a Dedekind domain, if \(\alpha \in K^*\) the fractional ideal

where the product is over all non-zero prime ideals \(\mathfrak {Q}\), all \(e_{\mathfrak {Q}}\in \mathbb {Z}\), and all but finitely many \(e_{\mathfrak {Q}}=0\). Then we define \(v_{\mathfrak {P}}(\alpha )=e_{\mathfrak {P}}\), and then the metric \(d\) on \(K\) by \(d(\alpha , \beta ) = p^{-v_{\mathfrak {P}}(\alpha -\beta )}\) if \(\alpha \neq \beta \) and \(d(\alpha ,\alpha )=0\). Then the completion of \(K\) with respect to this metric is \(K_{\mathfrak {P}}\).

If \(K=\mathbb {Q}(a)\), and \(f\in \mathbb {Q}[x]\) is the monic irreducible polynomial for \(a\) over \(\mathbb {Q}\), then adjoining the roots of \(f\) to \(\mathbb {Q}_p\) provide another means of constructing the completions.

Finally, the local algebra of \(K\), \(\prod _{j=1}^g K_j\) is a product of the \(p\)-adic completions of \(K\). The \(p\)-adic completions of \(K\) correspond to the nonarchimedian places of \(K\).

A place \(v\) of a field \(K\) is an equivalence class of non-trivial absolute values on \(K\). As with absolute values, places may be classified as archimedean or nonarchimedean, since these properties are preserved under equivalence.

Each place induces a distance metric that gives \(K\) a metric topology. The completion \(K_v\) of \(K\) at \(v\) is the completion of this metric space, which is also a topological field.

When \(K\) is a number field each nonarchimedean place arises from the valuation associated to each prime ideal in the ring of integers of \(K\), while archimedean places arise from embeddings of \(K\) into the complex numbers: each real embedding determines a real place, and each conjugate pair of complex embeddings determines a complex place. The archimedean places of a number field are also called infinite places.

Every number field \(K\) can be represented as \(K = \mathbb {Q}[X]/P(x)\) for some monic \(P\in \mathbb {Z}[X]\), called a defining polynomial for \(K\). Among all such defining polynomials, we define the reduced defining polynomial as follows.

Recall that for a monic polynomial \(P(x) = \prod _i(x-\alpha _i)\), the \(T_2\) norm of \(P\) is \(T_2(P) = \sum _i |\alpha _i|^2\).

• Let \(L_0\) be the list of (monic integral) defining polynomials for \(K\) that are minimal with respect to the \(T_2\) norm.

• Let \(L_1\) be the sublist of \(L_0\) of polynomials whose discriminant has minimal absolute value.

• For a polynomial \(P = x^n + a_1x^{n-1} + \dots + a_n\), let \(S(P) = (|a_1|,a_1,\dots ,|a_n|,a_n)\), and order the polynomials in \(L_1\) by the lexicographic order of the vectors \(S(P)\).

Then the reduced defining polynomial of \(K\) is the first polynomial in \(L_1\) with respect to this order.

The pari/gp function <code>polredabs()</code> computes reduced defining polynomials, which are also commonly called <code>polredabs</code> polynomials.

The discriminant of a monic polynomial \(f(x) = \prod _{i=1}^d (x - \alpha _i)\) is the quantity

If \(f\) has integral coefficients, \(K\) is the number field defined by \(f\) and \(\alpha \) is a root of \(f\) in \(K\), then the discriminant \(D\) of \(K\) divides \(\Delta \) and the ratio \(\Delta /D\) is the square of the index of \(\mathbb {Z}[\alpha ]\) in the ring of integers of \(K\).

A prime \(\mathfrak p\) of a number field \(K\) is a nonzero prime ideal of its ring of integers \(\mathcal O_K\).

The ideal \(\mathfrak p \cap \mathcal O_K\) is a nonzero prime ideal of \(\mathbb {Z}\) (a prime of \(\mathbb {Q}\)), which is necessarily a principal ideal \((p)\) for some prime number \(p\). The prime \(\mathfrak p\) is then said to be a prime above \(p\).

A prime integer \(p\) is a ramified prime of a number field \(K\) if, when the ideal generated by \(p\) is factored into prime ideals in the ring of integers \(\mathcal{O}_K\) of \(K\),

there is an \(i\) such that \(e_i\geq 2\).

The ramified primes of \(K\) are the primes dividing the discriminant of \(K\).

The rank of a number field \(K\) is the size of any set of fundamental units of \(K\). It is equal to \(r = r_1 + r_2 -1\) where \(r_1\) is the number of real embeddings of \(K\) into \(\mathbb {C}\) and \(2r_2\) is the number of complex embeddings of \(K\) into \(\mathbb {C}.\)

A real embedding of a number field \(K\) is a field homomorphism \(K\to \mathbb {R}\). A single number field may have several distinct real embeddings.

Let \(K\) be a CM number field and let \(\overline{\mathbb {Q}}\) be the algebraic closure of \(\mathbb {Q}\) in \(\mathbb {C}\). A subset \(\Phi \subset \mathrm{Hom}(K, \overline{\mathbb {Q}})\) is called a CM type if for every embedding \(\iota \in \mathrm{Hom}(K, \overline{\mathbb {Q}})\) either \(\iota \in \Phi \) or \(\overline{\iota } \in \Phi \), but not both, where \(\overline{\iota }\) is the complex conjugate of \(\iota \).

Given a CM field \(K\) and a CM type \(\Phi \), the reflex field is the fixed field inside \(\overline{\mathbb {Q}}\) corresponding to the subgroup \(\{ \rho \in \operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q}) : \rho \Phi = \Phi \} \) of \(\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\). A CM type \(\Phi \) and its complement \(\overline{\Phi }\), which is the same as the set of complex conjugate embeddings, have the same reflex field. The number of complex conjugate pairs of CM types is \(2^{g-1}\), where \(2g=[K:\mathbb {Q}]\), the degree of \(K\) over \(\mathbb {Q}\).

To specify a CM type \(\Phi \) for the CM field \(K=\mathbb {Q}(a)\): <ol> <li>fix an order \( (\iota _1,\overline{\iota _1}), \dots , (\iota _g,\overline{\iota _g}) \) of the pairs of complex embeddings of \(K\); <li> then \(\Phi =(\varphi _1,\dots ,\varphi _g)\) where \(\varphi _j\in \{ \iota _j,\overline{\iota _j}\} \) for \(1\le j\le g\); <li> now \(\Phi \) can be encoded by the list \((\text{sign}(\text{im}(\varphi _1(a))),\dots ,\text{sign}(\text{im}(\varphi _g(a))))\). </ol>

The CM types in the LMFDB are grouped in Galois orbits under the action of \(\mathrm{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\) described above.

<!— (commented out by John Cremona: this information should be in the completeness knowl for number fields) In the LMFDB, there is a potentially incomplete list of reflex fields for each CM field \(K\) of degree at most 12. For each reflex field, it is indicated for how many of the \(2^{[K:\mathbb {Q}]/2 - 1}\) pairs of complementary CM types this particular field is the reflex field. The only reflex fields listed are those of degree at most 36.

• –>

Let \(K\) be a CM number field and let \(N\) a normal closure of \(K\), let \(\Phi \subset \mathrm{Hom}(K, \overline{\mathbb {Q}})\) be a CM type and \(L\) its associated reflex field. Then \(\Phi \) induces a CM type \(\Phi _N \subset \mathrm{Hom}(N, \mathbb {C})\) by taking the maps that restrict to a map inside \(\Phi \) on \(K\). The maps in \(\Phi _N\) are isomorphisms on the image \(F\) of \(N\) inside \(\overline{\mathbb {Q}}\) and by inverting them, we obtain a CM type on \(F\) with values in \(N\). The reflex field of the reflex field is the reflex field of this CM type.

It can also be computed as follows. Consider the right action of \(\mathrm{Gal}(N/K)\) on the set of CM types on \(K\). Then the reflex field of the reflex field is the subfield corresponding to the subgroup stabilising \(\Phi \).

The reflex field of the reflex field is also the smallest field of definition of the CM type \(\Phi \), i.e. it is the largest subfield \(M\) of \(K\) such that \(\Phi \) is induced from a CM type on \(M\).

Let \(\sigma _1,\ldots ,\sigma _{r_1}\) be the real embeddings of a number field \(K\) into the complex numbers \(\mathbb {C}\), and \(\sigma _{r_1+ 1},\ldots ,\sigma _{r_1+r_2}\) be complex embeddings of \(K\) into \(\mathbb {C}\) such that no two are complex conjugate. Let \(u_1,\ldots ,u_r\) be a set of fundamental units of \(K\). Then \(r = r_1 + r_2 -1\).

Let \(M\) be the \((r_1+ r_2-1)\times (r_1+r_2)\) matrix \((d_j\log { \sigma _j(u_i)})\), where \(d_j=1\) if \(j\leq r_1\), i.e, if \(\sigma _j\) is a real embedding, and \(d_j=2\) otherwise, i.e., if \(\sigma _j\) is a complex embedding. The sum of the columns of \(M\) is the zero vector.

The regulator of \(K\) is the absolute value of the determinant of the sub-matrix of \(M\) where one column is removed. Its value is independent of the choice of column which is removed.

If \(K\) is a number field with CM with class number \(h\), and \(K^+\) is its maximal totally real subfield with class number \(h^+\), then \(h^+\) divides \(h\) and the relative class number is \(h/h^+\).

The ring of integers of a number field \(K\) is the integral closure of \(\mathbb {Z}\) in \(K\).

If \(K\) is a number field of degree \(n\) and discriminant \(D\), then the root discriminant of \(K\) is

It gives a measure of the discriminant of a number field which is normalized for the degree. For example, if \(K\subseteq L\) are number fields and \(L/K\) is unramified, then \(\textrm{rd}(K)=\textrm{rd}(L)\).

If \(K/F\) is a finite degree field extension, \(\alpha \in K\) is separable over \(F\) if its monic irreducible polynomial has distinct roots in the algebraic closure \(\overline{F}\).

The extension \(K/F\) is separable if every \(\alpha \in K\) is separable over \(F\).

All algebraic extensions of local and global number fields are separable.

A (finite) separable algebra \(A\) over a field \(F\), also called an étale \(F\)-algebra, is an \(F\)-algebra of finite dimension that is isomorphic to a product of separable field extensions of \(F\).

If \(L/K\) is a field extension and \(A\) is a separable \(K\)-algebra then \(A\otimes _K L\) is a separable \(L\)-algebra (which is typically not a field, even when \(A\) is).

For each positive integer \(n\), let \(C_n\) for the minimum root discriminant for all number fields of degree \(n\). Assuming the Generalized Riemann Hypothesis, \(\limsup C_n \geq \Omega \) where

and \(\gamma \) is the Euler–Mascheroni constant. Lower bounds for the \(C_n\) were deduced by analytic methods through the work of Odlyzko and others. In particular, Serre introduced the constant \(\Omega \) which we refer to as the Serre Odlyzko bound,

Consequently, any number field whose root discriminant lies below \(\Omega \) can be considered to have small discriminant.

Two finite separable extension fields \(K_1\) and \(K_2\) of a ground field \(F\) are called siblings if they are not isomorphic, but have isomorphic Galois closures.

A finite dimensional separable \(\mathbb {Q}\)-algebra is isomorphic to a product of number fields. By its Galois closure, we mean the compositum of the Galois closures of the constituent fields. Then two algebras are siblings if they have isomorphic Galois closures, but are not isomorphic as \(\mathbb {Q}\)-algebras.

The signature of a number field \(K\) is the pair \([r_1,r_2]\) where \(r_1\) is the number of real embeddings of \(K\) and \(r_2\) is the number of conjugate pairs of complex embeddings.

The degree of \(K\) is \(r_1+2r_2\).

If \(K/F\) is a Galois extension of fields, a stem field for \(K/F\) is a field \(E\) such that \(F\subseteq E\subseteq K\) and \(K\) is the Galois closure of \(E/F\).

This is connected to the notion of the stem field of a polynomial. If \(f\in F[x]\) is a separable irreducible polynomial of degree \(n\) with roots \(\alpha _1, \ldots , \alpha _n\) (in some extension field), then the fields \(F(\alpha _i)\) are the stem fields of the polynomial \(f\). The splitting field of \(f\) is \(K=F(\alpha _1,\ldots ,\alpha _n)\), which is a Galois extension of \(F\), and the fields \(F(\alpha _i)\) are stem fields for \(K/F\) as defined above.

A torsion generator of a number field is a primitive root of unity that generates the torsion subgroup of the unit group (which is necessarily cyclic).

A number field \(K\) is totally imaginary (or totally complex) if it cannot be embedded in the real numbers \(\mathbb {R}\); equivalently, \(\mathbb {R}\) does not contain the image of any of the homomorphisms from \(K\) to \(\mathbb {C}\).

An element \(\alpha \) in a number field \(K\) is totally positive if \(\sigma (\alpha ){\gt}0\) for all real embeddings \(\sigma \) of \(K\) into \(\mathbb {R}\).

A global number field \(K\) is always of the form \(K=\mathbb {Q}(\alpha )\) where \(\alpha \) has monic irreducible polynomial \(f(x)\in \mathbb {Q}[x]\). The field is totally real if all of the roots of \(f(x)\) in \(\mathbb {C}\) lie in the real numbers \(\mathbb {R}\).

Equivalently, \(K\) is totally real if all the embeddings of \(K\) into \(\mathbb {C}\) have image contained in \(\mathbb {R}\).

The unit group of a number field \(K\) is the group of units of the ring of integers of \(K\). It is a finitely generated abelian group with cyclic torsion subgroup. A set of generators of a maximal torsion-free subgroup is called a set of fundamental units for \(K\).

The unit group of \(K\) has as invariants the rank and the regulator of \(K\).

A unramified (rational) prime of a number field \(K\) is a prime integer \(p\) such that the ideal generated by \(p\) is factored into distinct prime ideals in the ring of integers \(\mathcal{O}_K\) of \(K\)

The unramified primes of \(K\) are the primes which do not divide the discriminant of \(K\).

The (logarithmic) Weil height of a nonzero rational number \(a/b\in \mathbb {Q}\) in lowest terms is the quantity

The height of \(0\) is taken to be \(0.\)

The (absolute logarithmic) Weil height of an element \(\alpha \) in a number field \(K\) is the quantity

where \(M_K\) is an appropriately normalized set of inequivalent absolute values on \(K\). More generally, the height of a point \(P=[\alpha _0,\alpha _1,\ldots ,\alpha _n]\) in projective space \(\mathbb {P}^n(K)\) is given by

If \(\mathcal{L}\) is a very amply line bundle on a projective variety \(V\) inducing an embedding \(\iota \colon V \hookrightarrow \mathbb {P}^n\), then the Weil height associated on \(X\) associated to \(\mathcal{L}\) is given by

This definition can be extended to all line bundles by using the following linearity:

For a prime power \(q\), a Weil \(q\)-polynomial is a monic polynomial with integer coefficients whose complex roots are of absolute value \(\sqrt{q}\).

Given \(q\) and a nonnegative integer \(d\), there are only finitely many Weil \(q\)-polynomials of degree \(d\).

The characteristic polynomial of an abelian variety over \(\mathbb {F}_q\) is a Weil \(q\)-polynomial, but it is not quite true that every Weil \(q\)-polynomial arises in this way. Every irreducible Weil \(q\)-polynomial has a unique power that is the characteristic polynomial of a simple abelian variety over \(\mathbb {F}_q\); it is the products of these powers that arise from abelian varieties.

If \(K\) is a number field with ring of integers \(\mathcal{O}_K\), then for all \(\alpha \in \mathcal{O}_K\) such that \(K=\mathbb {Q}(\alpha )\), the index of \(\alpha \), \(i(\alpha )\) is the index of the order \(\mathbb {Z}[\alpha ]\).

The index of the number field is the greatest common divisor of all \(i(\alpha )\) with \(\alpha \) as above.