Number field operations

Creation of number fields

General number fields can be created using the function number_field. To create a simple number field given by a defining polynomial or a non-simple number field given by defining polynomials, the following functions can be used.

number_field Method

number_field(f::Poly{NumFieldElem}, s::VarName;
            cached::Bool = false, check::Bool = false) -> NumField, NumFieldElem

Given an irreducible polynomial $f\in K[x]$ over some number field $K$, this function creates the simple number field $L=K[x]/(f)$ and returns $(L,b)$, where $b$ is the class of $x$ in $L$. The string s is used only for printing the primitive element $b$.

• check: Controls whether irreducibility of $f$ is checked.

• cached: Controls whether the result is cached.

Examples

julia> K, a = quadratic_field(5);

julia> Kt, t = K["t"];

julia> L, b = number_field(t^3 - 3, "b");

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number_field Method

number_field(f::Vector{PolyRingElem{<:NumFieldElem}}, s::VarName="_\$", check = true)
                                          -> NumField, Vector{NumFieldElem}

Given a list $f_1,\dots ,f_n$ of univariate polynomials in $K[x]$ over some number field $K$, constructs the extension $K[x_1,\dots ,x_n]/(f_1(x_1),\dots ,f_n(x_n))$.

Examples

julia> Qx, x = QQ["x"];

julia> K, a = number_field([x^2 - 2, x^2 - 3], "a")
(Non-simple number field of degree 4 over QQ, AbsNonSimpleNumFieldElem[a1, a2])

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Tip

Many of the constructors have arguments of type Symbol or AbstractString. If used, they define the appearance in printing, and printing only. The named parameter check can be true or false, the default being true. This parameter controls whether the polynomials defining the number field are tested for irreducibility or not. Given that this can be potentially very time consuming if the degree if large, one can disable this test. Note however, that the behaviour of Hecke is undefined if a reducible polynomial is used to define a field.

The named boolean parameter cached can be used to disable caching. Two number fields defined using the same polynomial from the identical polynomial ring and the same (identical) symbol/string will be identical if cached == true and different if cached == false.

For frequently used number fields like quadratic fields, cyclotomic fields or radical extensions, the following functions are provided:

cyclotomic_field Method

cyclotomic_field(n::Int, s::VarName = "z_$n", t = "_\$"; cached::Bool = true)

Return a tuple $R,x$ consisting of the parent object $R$ and generator $x$ of the $n$-th cyclotomic field, $\mathbb{Q}({\zeta }_n)$. The supplied string s specifies how the generator of the number field should be printed. If provided, the string t specifies how the generator of the polynomial ring from which the number field is constructed, should be printed. If it is not supplied, a default dollar sign will be used to represent the variable.

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quadratic_field Method

quadratic_field(d::IntegerUnion) -> AbsSimpleNumField, AbsSimpleNumFieldElem

Returns the field with defining polynomial $x^2-d$.

Examples

julia> quadratic_field(5)
(Real quadratic field defined by x^2 - 5, sqrt(5))

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wildanger_field Method

wildanger_field(n::Int, B::ZZRingElem) -> AbsSimpleNumField, AbsSimpleNumFieldElem

Returns the field with defining polynomial $x^n+\sum _{i=0}^{n-1}(-1{)}^{n-i}B{x}^i$. These fields tend to have non-trivial class groups.

Examples

julia> wildanger_field(3, ZZ(10), "a")
(Number field of degree 3 over QQ, a)

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radical_extension Method

radical_extension(n::Int, a::NumFieldElem, s = "_$";
               check = true, cached = true) -> NumField, NumFieldElem

Given an element $a$ of a number field $K$ and an integer $n$, create the simple extension of $K$ with the defining polynomial $x^n-a$.

Examples

julia> radical_extension(5, QQ(2), "a")
(Number field of degree 5 over QQ, a)

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rationals_as_number_field Method

rationals_as_number_field() -> AbsSimpleNumField, AbsSimpleNumFieldElem

Returns the rational numbers as the number field defined by $x-1$.

Examples

julia> rationals_as_number_field()
(Number field of degree 1 over QQ, 1)

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Basic properties

basis Method

basis(L::SimpleNumField) -> Vector{NumFieldElem}

Return the canonical basis of a simple extension $L/K$, that is, the elements $1,a,\dots ,a^{d-1}$, where $d$ is the degree of $K$ and $a$ the primitive element.

Examples

julia> Qx, x = QQ["x"];

julia> K, a = number_field(x^2 - 2, "a");

julia> basis(K)
2-element Vector{AbsSimpleNumFieldElem}:
 1
 a

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basis Method

basis(L::NonSimpleNumField) -> Vector{NumFieldElem}

Returns the canonical basis of a non-simple extension $L/K$. If $L=K(a_1,\dots ,a_n)$ where each $a_i$ has degree $d_i$, then the basis will be $a_1^{i_1}\cdots a_d^{i_d}$ with $0\le i_j\le d_j-1$ for $1\le j\le n$.

Examples

julia> Qx, x = QQ["x"];

julia> K, (a1, a2) = number_field([x^2 - 2, x^2 - 3], "a");

julia> basis(K)
4-element Vector{AbsNonSimpleNumFieldElem}:
 1
 a1
 a2
 a1*a2

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absolute_basis Method

absolute_basis(K::NumField) -> Vector{NumFieldElem}

Returns an array of elements that form a basis of $K$ (as a vector space) over the rationals.

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defining_polynomial Method

defining_polynomial(L::SimpleNumField) -> PolyRingElem

Given a simple number field $L/K$, constructed as $L=K[x]/(f)$, this function returns $f$.

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defining_polynomials Method

defining_polynomials(L::NonSimpleNumField) -> Vector{PolyRingElem}

Given a non-simple number field $L/K$, constructed as $L=K[x]/(f_1,\dots ,f_r)$, return the vector containing the $f_i$'s.

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absolute_primitive_element Method

absolute_primitive_element(K::NumField) -> NumFieldElem

Given a number field $K$, this function returns an element $\gamma \in K$ such that $K=\mathbf{Q}(\gamma )$.

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component Method

component(L::NonSimpleNumField, i::Int) -> SimpleNumField, Map

Given a non-simple extension $L/K$, this function returns the simple number field corresponding to the $i$-th component of $L$ together with its embedding.

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base_field Method

base_field(L::NumField) -> NumField

Given a number field $L/K$ this function returns the base field $K$. For absolute extensions this returns $\mathbf{Q}$.

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Invariants

degree Method

degree(L::NumField) -> Int

Given a number field $L/K$, this function returns the degree of $L$ over $K$.

Examples

julia> Qx, x = QQ["x"];

julia> K, a = number_field(x^2 - 2, "a");

julia> degree(K)
2

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absolute_degree Method

absolute_degree(L::NumField) -> Int

Given a number field $L/K$, this function returns the degree of $L$ over $\mathbf{Q}$.

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signature Method

signature(K::NumField)

Return the signature of the number field of $K$.

Examples

julia> Qx, x = QQ["x"];

julia> K, a = number_field(x^2 - 2, "a");

julia> signature(K)
(2, 0)

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unit_group_rank Method

unit_group_rank(K::NumField) -> Int

Return the rank of the unit group of any order of $K$.

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class_number Method

class_number(K::AbsSimpleNumField) -> ZZRingElem

Returns the class number of $K$.

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relative_class_number Method

relative_class_number(K::AbsSimpleNumField) -> ZZRingElem

Returns the relative class number of $K$. The field must be a CM-field.

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regulator Method

regulator(K::AbsSimpleNumField)

Computes the regulator of $K$, i.e. the discriminant of the unit lattice for the maximal order of $K$.

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discriminant Method

discriminant(L::SimpleNumField) -> NumFieldElem

The discriminant of the defining polynomial of $L$, not the discriminant of the maximal order of $L$.

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absolute_discriminant Method

absolute_discriminant(L::SimpleNumField, QQ) -> QQFieldElem

The absolute discriminant of the defining polynomial of $L$, not the discriminant of the maximal order of $L$. This is the norm of the discriminant times the $d$-th power of the discriminant of the base field, where $d$ is the degree of $L$.

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Predicates

is_simple Method

is_simple(L::NumField) -> Bool

Given a number field $L/K$ this function returns whether $L$ is simple, that is, whether $L/K$ is defined by a univariate polynomial.

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is_absolute Method

is_absolute(L::NumField) -> Bool

Returns whether $L$ is an absolute extension, that is, whether the base field of $L$ is $\mathbf{Q}$.

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is_totally_real Method

is_totally_real(K::NumField) -> Bool

Return true if and only if $K$ is totally real, that is, if all roots of the defining polynomial are real.

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is_totally_complex Method

is_totally_complex(K::NumField) -> Bool

Return true if and only if $K$ is totally complex, that is, if all roots of the defining polynomial are not real.

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is_cm_field Method

is_cm_field(K::AbsSimpleNumField) -> Bool, NumFieldHom{AbsSimpleNumField, AbsSimpleNumField}

Given a number field $K$, this function returns true and the complex conjugation if the field is CM, false and the identity otherwise.

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is_kummer_extension Method

is_kummer_extension(L::SimpleNumField) -> Bool

Tests if $L/K$ is a Kummer extension, that is, if the defining polynomial is of the form $x^n-b$ for some $b\in K$ and if $K$ contains the $n$-th roots of unity.

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is_radical_extension Method

is_radical_extension(L::SimpleNumField) -> Bool

Tests if $L/K$ is pure, that is, if the defining polynomial is of the form $x^n-b$ for some $b\in K$.

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is_linearly_disjoint Method

is_linearly_disjoint(K::SimpleNumField, L::SimpleNumField) -> Bool

Given two number fields $K$ and $L$ with the same base field $k$, this function returns whether $K$ and $L$ are linear disjoint over $k$.

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is_weakly_ramified Method

is_weakly_ramified(K::AbsSimpleNumField, P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Bool

Given a prime ideal $P$ of a number field $K$, return whether $P$ is weakly ramified, that is, whether the second ramification group is trivial.

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is_tamely_ramified Method

is_tamely_ramified(K::AbsSimpleNumField) -> Bool

Returns whether the number field $K$ is tamely ramified.

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is_tamely_ramified Method

is_tamely_ramified(O::AbsSimpleNumFieldOrder, p::Union{Int, ZZRingElem}) -> Bool

Returns whether the integer $p$ is tamely ramified in $\mathcal{O}$. It is assumed that $p$ is prime.

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is_abelian Method

is_abelian(L::NumField) -> Bool

Check if the number field $L/K$ is abelian over $K$. The function is probabilistic and assumes GRH.

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Subfields

is_subfield Method

is_subfield(K::SimpleNumField, L::SimpleNumField) -> Bool, Map

Return true and an injection from $K$ to $L$ if $K$ is a subfield of $L$. Otherwise the function returns false and a morphism mapping everything to $0$.

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subfields Method

subfields(L::SimpleNumField) -> Vector{Tuple{NumField, Map}}

Given a simple extension $L/K$, returns all subfields of $L$ containing $K$ as tuples $(k,\iota )$ consisting of a simple extension $k$ and an embedding $\iota k\to K$.

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principal_subfields Method

principal_subfields(L::SimpleNumField) -> Vector{Tuple{NumField, Map}}

Return the principal subfields of $L$ as pairs consisting of a subfield $k$ and an embedding $k\to L$.

Examples

julia> Qx, x = QQ["x"];

julia> K, a = number_field(x^8 - x^4 + 1);

julia> length(principal_subfields(K))
8

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compositum Method

compositum(K::AbsSimpleNumField, L::AbsSimpleNumField) -> AbsSimpleNumField, Map, Map

Assuming $L$ is normal (which is not checked), compute the compositum $C$ of the 2 fields together with the embedding of $K\to C$ and $L\to C$.

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embedding Method

embedding(k::NumField, K::NumField) -> Map

Assuming $k$ is known to be a subfield of $K$, return the embedding map.

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normal_closure Method

normal_closure(K::AbsSimpleNumField) -> AbsSimpleNumField, NumFieldHom{AbsSimpleNumField, AbsSimpleNumField}

The normal closure of $K$ together with the embedding map.

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relative_simple_extension Method

relative_simple_extension(K::NumField, k::NumField) -> RelSimpleNumField

Given two fields $K\supset k$, it returns $K$ as a simple relative extension $L$ of $k$ and an isomorphism $L\to K$.

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is_subfield_normal Method

  is_subfield_normal(K::AbsSimpleNumField, L::AbsSimpleNumField) -> Bool, NumFieldHom{AbsSimpleNumField, AbsSimpleNumField}

Returns true and an injection from $K$ to $L$ if $K$ is a subfield of $L$. Otherwise the function returns false and a morphism mapping everything to 0.

This function assumes that $K$ is normal.

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Conversion

simplify Method

simplify(K::AbsSimpleNumField; canonical::Bool = false) -> AbsSimpleNumField, NumFieldHom{AbsSimpleNumField, AbsSimpleNumField}

Tries to find an isomorphic field $L$ given by a "simpler" defining polynomial. By default, "simple" is defined to be of smaller index, testing is done only using a LLL-basis of the maximal order.

If canonical is set to true, then a canonical defining polynomial is found, where canonical is using the definition of PARI's polredabs, which is described in http://beta.lmfdb.org/knowledge/show/nf.polredabs.

Both versions require a LLL reduced basis for the maximal order.

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absolute_simple_field Method

absolute_simple_field(K::NumField) -> NumField, Map

Given a number field $K$, this function returns an absolute simple number field $M/\mathbf{Q}$ together with a $\mathbf{Q}$-linear isomorphism $M\to K$.

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simple_extension Method

simple_extension(L::NonSimpleNumField) -> SimpleNumField, Map

Given a non-simple extension $L/K$, this function computes a simple extension $M/K$ and a $K$-linear isomorphism $M\to L$.

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simplified_simple_extension Method

simplified_simple_extension(L::NonSimpleNumField) -> SimpleNumField, Map

Given a non-simple extension $L/K$, this function returns an isomorphic simple number field with a "small" defining equation together with the isomorphism.

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Morphisms

is_isomorphic Method

is_isomorphic(K::SimpleNumField, L::SimpleNumField) -> Bool

Return true if $K$ and $L$ are isomorphic, otherwise false.

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is_isomorphic_with_map Method

is_isomorphic_with_map(K::SimpleNumField, L::SimpleNumField) -> Bool, Map

Return true and an isomorphism from $K$ to $L$ if $K$ and $L$ are isomorphic. Otherwise the function returns false and a morphism mapping everything to $0$.

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is_involution Method

is_involution(f::NumFieldHom{AbsSimpleNumField, AbsSimpleNumField}) -> Bool

Returns true if $f$ is an involution, i.e. if $f^2$ is the identity, false otherwise.

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fixed_field Method

fixed_field(K::SimpleNumField,
            sigma::Map;
            simplify::Bool = true) -> number_field, NumFieldHom{AbsSimpleNumField, AbsSimpleNumField}

Given a number field $K$ and an automorphism $\sigma$ of $K$, this function returns the fixed field of $\sigma$ as a pair $(L,i)$ consisting of a number field $L$ and an embedding of $L$ into $K$.

By default, the function tries to find a small defining polynomial of $L$. This can be disabled by setting simplify = false.

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automorphism_list Method

automorphism_list(L::NumField) -> Vector{NumFieldHom}

Given a number field $L/K$, return a list of all $K$-automorphisms of $L$.

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automorphism_group Method

automorphism_group(K::NumField) -> GenGrp, GrpGenToNfMorSet

Given a number field $K$, this function returns a group $G$ and a map from $G$ to the automorphisms of $K$.

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complex_conjugation Method

complex_conjugation(K::AbsSimpleNumField)

Given a normal number field, this function returns an automorphism which is the restriction of complex conjugation at one embedding.

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Galois theory

normal_basis Method

normal_basis(L::NumField) -> NumFieldElem

Given a normal number field $L/K$, this function returns an element $a$ of $L$, such that the orbit of $a$ under the Galois group of $L/K$ is an $K$-basis of $L$.

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decomposition_group Method

decomposition_group(K::AbsSimpleNumField, P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, m::Map)
                                              -> Grp, GrpToGrp

Given a prime ideal $P$ of a number field $K$ and a map m return from automorphism_group(K), return the decomposition group of $P$ as a subgroup of the domain of m.

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ramification_group Method

ramification_group(K::AbsSimpleNumField, P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, m::Map) -> Grp, GrpToGrp

Given a prime ideal $P$ of a number field $K$ and a map m return from automorphism_group(K), return the ramification group of $P$ as a subgroup of the domain of m.

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inertia_subgroup Method

inertia_subgroup(K::AbsSimpleNumField, P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, m::Map) -> Grp, GrpToGrp

Given a prime ideal $P$ of a number field $K$ and a map m return from automorphism_group(K), return the inertia subgroup of $P$ as a subgroup of the domain of m.

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Infinite places

infinite_places Method

infinite_places(K::NumField) -> Vector{InfPlc}

Return all infinite places of the number field.

Examples

julia> K,  = quadratic_field(5);

julia> infinite_places(K)
2-element Vector{InfPlc{AbsSimpleNumField, AbsSimpleNumFieldEmbedding}}:
 Infinite place corresponding to (Complex embedding corresponding to -2.24 of real quadratic field)
 Infinite place corresponding to (Complex embedding corresponding to 2.24 of real quadratic field)

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real_places Method

real_places(K::NumField) -> Vector{InfPlc}

Return all infinite real places of the number field.

Examples

julia> K,  = quadratic_field(5);

julia> infinite_places(K)
2-element Vector{InfPlc{AbsSimpleNumField, AbsSimpleNumFieldEmbedding}}:
 Infinite place corresponding to (Complex embedding corresponding to -2.24 of real quadratic field)
 Infinite place corresponding to (Complex embedding corresponding to 2.24 of real quadratic field)

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complex_places Method

complex_places(K::NumField) -> Vector{InfPlc}

Return all infinite complex places of $K$.

Examples

julia> K,  = quadratic_field(-5);

julia> complex_places(K)
1-element Vector{InfPlc{AbsSimpleNumField, AbsSimpleNumFieldEmbedding}}:
 Infinite place corresponding to (Complex embedding corresponding to 0.00 + 2.24 * i of imaginary quadratic field)

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isreal Method

isreal(P::Plc)

Return whether the embedding into $\mathbf{C}$ defined by $P$ is real or not.

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is_complex Method

is_complex(P::Plc) -> Bool

Return whether the embedding into $\mathbf{C}$ defined by $P$ is complex or not.

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Miscellaneous

norm_equation Method

norm_equation(K::AnticNumerField, a) -> AbsSimpleNumFieldElem

For $a$ an integer or rational, try to find $T\in K$ s.th. $N(T)=a$. Raises an error if unsuccessful.

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lorenz_module Method

lorenz_module(k::AbsSimpleNumField, n::Int) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}

Finds an ideal $A$ s.th. for all positive units $e=1modA$ we have that $e$ is an $n$-th power. Uses Lorenz, number theory, 9.3.1. If containing is set, it has to be an integral ideal. The resulting ideal will be a multiple of this.

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kummer_failure Method

kummer_failure(x::AbsSimpleNumFieldElem, M::Int, N::Int) -> Int

Computes the quotient of $N$ and $[K({\zeta }_M,\sqrt[N]{(}x)):K({\zeta }_M)]$, where $K$ is the field containing $x$ and $N$ divides $M$.

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is_defining_polynomial_nice Method

is_defining_polynomial_nice(K::AbsSimpleNumField)

Tests if the defining polynomial of $K$ is integral and monic.

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