Spaces

Creation of spaces

# quadratic_space — Method.

quadratic_space(K::NumField, n::Int; cached::Bool = true) -> QuadSpace

Create the quadratic space over K with dimension n and Gram matrix equals to the identity matrix.

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# hermitian_space — Method.

hermitian_space(E::NumField, n::Int; cached::Bool = true) -> HermSpace

Create the hermitian space over E with dimension n and Gram matrix equals to the identity matrix. The number field E must be a quadratic extension, that is, $degree(E)==2$ must hold.

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# quadratic_space — Method.

quadratic_space(K::NumField, G::MatElem; cached::Bool = true) -> QuadSpace

Create the quadratic space over K with Gram matrix G. The matrix G must be square and symmetric.

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# hermitian_space — Method.

hermitian_space(E::NumField, gram::MatElem; cached::Bool = true) -> HermSpace

Create the hermitian space over E with Gram matrix equals to gram. The matrix gram must be square and hermitian with respect to the non-trivial automorphism of E. The number field E must be a quadratic extension, that is, $degree(E)==2$ must hold.

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Examples

Here are easy examples to see how these constructors work. We will keep the two following spaces for the rest of this section:

julia> K, a = cyclotomic_real_subfield(7);

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

julia> E, b = number_field(t^2-a*t+1, "b");

julia> Q = quadratic_space(K, K[0 1; 1 0])
Quadratic space of dimension 2
  over maximal real subfield of cyclotomic field of order 7
with gram matrix
[0   1]
[1   0]

julia> H = hermitian_space(E, 3)
Hermitian space of dimension 3
  over relative number field with defining polynomial t^2 - (z_7 + 1/z_7)*t + 1
    over number field with defining polynomial $^3 + $^2 - 2*$ - 1
      over rational field
with gram matrix
[1   0   0]
[0   1   0]
[0   0   1]

---

Attributes

Let $(V,\mathrm{\Phi })$ be a space over $E/K$. We define its dimension to be its dimension as a vector space over its base ring $E$ and its rank to be the rank of its Gram matrix. If these two invariants agree, the space is said to be regular.

While dealing with lattices, one always works with regular ambient spaces.

The determinant $\text{det}(V,\mathrm{\Phi })$ of $(V,\mathrm{\Phi })$ is defined to be the class of the determinant of its Gram matrix in $K^{\times }/N(E^{\times })$ (which is similar to $K^{\times }/(K^{\times }{)}^2$ in the quadratic case). The discriminant $\text{disc}(V,\mathrm{\Phi })$ of $(V,\mathrm{\Phi })$ is defined to be $(-1{)}^{(m(m-1)/2)}\text{det}(V,\mathrm{\Phi })$, where $m$ is the rank of $(V,\mathrm{\Phi })$.

# rank — Method.

rank(V::AbstractSpace) -> Int

Return the rank of the space V.

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# dim — Method.

dim(V::AbstractSpace) -> Int

Return the dimension of the space V.

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# gram_matrix — Method.

gram_matrix(V::AbstractSpace) -> MatElem

Return the Gram matrix of the space V.

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# involution — Method.

involution(V::AbstractSpace) -> NumFieldHom

Return the involution of the space V.

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# base_ring — Method.

base_ring(V::AbstractSpace) -> NumField

Return the algebra over which the space V is defined.

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# fixed_field — Method.

fixed_field(V::AbstractSpace) -> NumField

Return the fixed field of the space V.

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# det — Method.

det(V::AbstractSpace) -> FieldElem

Return the determinant of the space V as an element of its fixed field.

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# discriminant — Method.

discriminant(V::AbstractSpace) -> FieldElem

Return the discriminant of the space V as an element of its fixed field.

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Examples

So for instance, one could get the following information about the hermitian space $H$:

julia> K, a = cyclotomic_real_subfield(7);

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

julia> E, b = number_field(t^2-a*t+1, "b");

julia> H = hermitian_space(E, 3);

julia> rank(H), dim(H)
(3, 3)

julia> gram_matrix(H)
[1   0   0]
[0   1   0]
[0   0   1]

julia> involution(H)
Map
  from relative number field of degree 2 over maximal real subfield of cyclotomic field of order 7
  to relative number field of degree 2 over maximal real subfield of cyclotomic field of order 7

julia> base_ring(H)
Relative number field with defining polynomial t^2 - (z_7 + 1/z_7)*t + 1
  over number field with defining polynomial $^3 + $^2 - 2*$ - 1
    over rational field

julia> fixed_field(H)
Number field with defining polynomial $^3 + $^2 - 2*$ - 1
  over rational field

julia> det(H), discriminant(H)
(1, -1)

---

Predicates

Let $(V,\mathrm{\Phi })$ be a hermitian space over $E/K$ (resp. quadratic space $K$). We say that $(V,\mathrm{\Phi })$ is definite if $E/K$ is CM (resp. $K$ is totally real) and if there exists an orthogonal basis of $V$ for which the diagonal elements of the associated Gram matrix of $(V,\mathrm{\Phi })$ are either all totally positive or all totally negative. In the former case, $V$ is said to be positive definite, while in the latter case it is negative definite. In all the other cases, we say that $V$ is indefinite.

# is_regular — Method.

is_regular(V::AbstractSpace) -> Bool

Return whether the space V is regular, that is, if the Gram matrix has full rank.

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# is_quadratic — Method.

is_quadratic(V::AbstractSpace) -> Bool

Return whether the space V is quadratic.

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# is_hermitian — Method.

is_hermitian(V::AbstractSpace) -> Bool

Return whether the space V is hermitian.

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# is_positive_definite — Method.

is_positive_definite(V::AbstractSpace) -> Bool

Return whether the space V is positive definite.

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# is_negative_definite — Method.

is_negative_definite(V::AbstractSpace) -> Bool

Return whether the space V is negative definite.

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# is_definite — Method.

is_definite(V::AbstractSpace) -> Bool

Return whether the space V is definite.

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Note that the is_hermitian function tests whether the space is non-quadratic.

Examples

julia> K, a = cyclotomic_real_subfield(7);

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

julia> E, b = number_field(t^2-a*t+1, "b");

julia> Q = quadratic_space(K, K[0 1; 1 0]);

julia> H = hermitian_space(E, 3);

julia> is_regular(Q), is_regular(H)
(true, true)

julia> is_quadratic(Q), is_hermitian(H)
(true, true)

julia> is_definite(Q), is_positive_definite(H)
(false, true)

---

Inner products and diagonalization

# gram_matrix — Method.

gram_matrix(V::AbstractSpace, M::MatElem) -> MatElem

Return the Gram matrix of the rows of M with respect to the Gram matrix of the space V.

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# gram_matrix — Method.

gram_matrix(V::AbstractSpace, S::Vector{Vector}) -> MatElem

Return the Gram matrix of the sequence S with respect to the Gram matrix of the space V.

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# inner_product — Method.

inner_product(V::AbstractSpace, v::Vector, w::Vector) -> FieldElem

Return the inner product of v and w with respect to the bilinear form of the space V.

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# orthogonal_basis — Method.

orthogonal_basis(V::AbstractSpace) -> MatElem

Return a matrix M, such that the rows of M form an orthogonal basis of the space V.

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# diagonal — Method.

diagonal(V::AbstractSpace) -> Vector{FieldElem}

Return a vector of elements $a_1,\dots ,a_n$ such that the space V is isometric to the diagonal space $⟨a_1,\dots ,a_n⟩$.

The elements are contained in the fixed field of V.

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# diagonal_with_transform — Method.

diagonal_with_transform(V::AbstractSpace) -> Vector{FieldElem},
                                                         MatElem{FieldElem}

Return a vector of elements $a_1,\dots ,a_n$ such that the space V is isometric to the diagonal space $⟨a_1,\dots ,a_n⟩$. The second output is a matrix U whose rows span an orthogonal basis of V for which the Gram matrix is given by the diagonal matrix of the $a_i$'s.

The elements are contained in the fixed field of V.

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# restrict_scalars — Method.

restrict_scalars(V::AbstractSpace, K::QQField,
                                   alpha::FieldElem = one(base_ring(V)))
                                                -> QuadSpace, AbstractSpaceRes

Given a space $(V,\mathrm{\Phi })$ and a subfield K of the base algebra E of V, return the quadratic space W obtained by restricting the scalars of $(V,\alpha \mathrm{\Phi })$ to K, together with the map f for extending the scalars back. The form on the restriction is given by $Tr\circ \mathrm{\Phi }$ where $Tr:E\to K$ is the trace form. The rescaling factor $\alpha$ is set to 1 by default.

Note that for now one can only restrict scalars to $\mathbb{Q}$.

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Examples

julia> K, a = cyclotomic_real_subfield(7);

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

julia> E, b = number_field(t^2-a*t+1, "b");

julia> Q = quadratic_space(K, K[0 1; 1 0]);

julia> H = hermitian_space(E, 3);

julia> gram_matrix(Q, K[1 1; 2 0])
[2   2]
[2   0]

julia> gram_matrix(H, E[1 0 0; 0 1 0; 0 0 1])
[1   0   0]
[0   1   0]
[0   0   1]

julia> inner_product(Q, K[1  1], K[0  2])
[2]

julia> orthogonal_basis(H)
[1   0   0]
[0   1   0]
[0   0   1]

julia> diagonal(Q), diagonal(H)
(AbsSimpleNumFieldElem[1, -1], AbsSimpleNumFieldElem[1, 1, 1])

---

Equivalence

Let $(V,\mathrm{\Phi })$ and $(V^{\prime },{\mathrm{\Phi }}^{\prime })$ be spaces over the same extension $E/K$. A homomorphism of spaces from $V$ to $V^{\prime }$ is a $E$-linear mapping $f:V\to V^{\prime }$ such that for all $x,y\in V$, one has

$${\mathrm{\Phi }}^{\prime }(f(x),f(y))=\mathrm{\Phi }(x,y).$$

An automorphism of spaces is called an isometry and a monomorphism is called an embedding.

# hasse_invariant — Method.

hasse_invariant(V::QuadSpace, p::Union{InfPlc, AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> Int

Returns the Hasse invariant of the quadratic space V at p. This is equal to the product of local Hilbert symbols $(a_i,a_j{)}_p$, $i<j$, where $V$ is isometric to $⟨a_1,\dots ,a_n⟩$. If V is degenerate return the hasse invariant of V/radical(V).

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# witt_invariant — Method.

witt_invariant(V::QuadSpace, p::Union{InfPlc, AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> Int

Returns the Witt invariant of the quadratic space V at p.

See [Definition 3.2.1, Kir16].

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# is_isometric — Method.

is_isometric(L::AbstractSpace, M::AbstractSpace) -> Bool

Return whether the spaces L and M are isometric.

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# is_isometric — Method.

is_isometric(L::AbstractSpace, M::AbstractSpace, p::Union{InfPlc, AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> Bool

Return whether the spaces L and M are isometric over the completion at p.

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# invariants — Method.

invariants(M::QuadSpace)
      -> FieldElem, Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Int}, Vector{Tuple{InfPlc, Int}}

Returns a tuple (n, k, d, H, I) of invariants of M, which determine the isometry class completely. Here n is the dimension. The dimension of the kernel is k. The element d is the determinant of a Gram matrix of the non-degenerate part, H contains the non-trivial Hasse invariants and I contains for each real place the negative index of inertia.

Note that d is determined only modulo squares.

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Examples

For instance, for the case of $Q$ and the totally ramified prime $\mathfrak{p}$ of $O_K$ above $7$, one can get:

julia> K, a = cyclotomic_real_subfield(7);

julia> Q = quadratic_space(K, K[0 1; 1 0]);

julia> OK = maximal_order(K);

julia> p = prime_decomposition(OK, 7)[1][1];

julia> hasse_invariant(Q, p), witt_invariant(Q, p)
(1, 1)

julia> Q2 = quadratic_space(K, K[-1 0; 0 1]);

julia> is_isometric(Q, Q2, p)
true

julia> is_isometric(Q, Q2)
true

julia> invariants(Q2)
(2, 0, -1, Dict{AbsSimpleNumFieldOrderIdeal, Int64}(), Tuple{InfPlc{AbsSimpleNumField, AbsSimpleNumFieldEmbedding}, Int64}[(Infinite place corresponding to (Complex embedding corresponding to -1.80 of maximal real subfield of cyclotomic field of order 7), 1), (Infinite place corresponding to (Complex embedding corresponding to -0.45 of maximal real subfield of cyclotomic field of order 7), 1), (Infinite place corresponding to (Complex embedding corresponding to 1.25 of maximal real subfield of cyclotomic field of order 7), 1)])

---

Embeddings

Let $(V,\mathrm{\Phi })$ and $(V^{\prime },{\mathrm{\Phi }}^{\prime })$ be two spaces over the same extension $E/K$, and let $\sigma :V\to V^{\prime }$ be an $E$-linear morphism. $\sigma$ is called a representation of $V$ into $V^{\prime }$ if for all $x\in V$

$${\mathrm{\Phi }}^{\prime }(\sigma (x),\sigma (x))=\mathrm{\Phi }(x,x).$$

In such a case, $V$ is said to be represented by $V^{\prime }$ and $\sigma$ can be seen as an embedding of $V$ into $V^{\prime }$. This representation property can be also tested locally with respect to the completions at some finite places. Note that in both quadratic and hermitian cases, completions are taken at finite places of the fixed field $K$.

# is_locally_represented_by — Method.

is_locally_represented_by(U::T, V::T, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) where T <: AbstractSpace -> Bool

Given two spaces U and V over the same algebra E, and a prime ideal p in the maximal order ${\mathcal{O}}_K$ of their fixed field K, return whether U is represented by V locally at p, i.e. whether $U_p$ embeds in $V_p$.

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# is_represented_by — Method.

is_represented_by(U::T, V::T) where T <: AbstractSpace -> Bool

Given two spaces U and V over the same algebra E, return whether U is represented by V, i.e. whether U embeds in V.

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Examples

Still using the spaces $Q$ and $H$, we can decide whether some other spaces embed respectively locally or globally into $Q$ or $H$:

julia> K, a = cyclotomic_real_subfield(7);

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

julia> E, b = number_field(t^2-a*t+1, "b");

julia> Q = quadratic_space(K, K[0 1; 1 0]);

julia> H = hermitian_space(E, 3);

julia> OK = maximal_order(K);

julia> p = prime_decomposition(OK, 7)[1][1];

julia> Q2 = quadratic_space(K, K[-1 0; 0 1]);

julia> H2 = hermitian_space(E, E[-1 0 0; 0 1 0; 0 0 -1]);

julia> is_locally_represented_by(Q2, Q, p)
true

julia> is_represented_by(Q2, Q)
true

julia> is_locally_represented_by(H2, H, p)
true

julia> is_represented_by(H2, H)
false

---

Categorical constructions

One can construct direct sums of spaces of the same kind. Since those are also direct products, they are called biproducts in this context. Depending on the user usage, one of the following three methods can be called to obtain the direct sum of a finite collection of spaces. Note that the corresponding copies of the original spaces in the direct sum are pairwise orthogonal.

# direct_sum — Method.

direct_sum(x::Vararg{T}) where T <: AbstractSpace -> T, Vector{AbstractSpaceMor}
direct_sum(x::Vector{T}) where T <: AbstractSpace -> T, Vector{AbstractSpaceMor}

Given a collection of quadratic or hermitian spaces $V_1,\dots ,V_n$, return their direct sum $V:=V_1\oplus \dots \oplus V_n$, together with the injections $V_i\to V$.

For objects of type AbstractSpace, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain V as a direct product with the projections $V\to V_i$, one should call direct_product(x). If one wants to obtain V as a biproduct with the injections $V_i\to V$ and the projections $V\to V_i$, one should call biproduct(x).

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direct_sum(g1::QuadSpaceCls, g2::QuadSpaceCls) -> QuadSpaceCls

Return the isometry class of the direct sum of two representatives.

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# direct_product — Method.

direct_product(algebras::StructureConstantAlgebra...; task::Symbol = :sum)
  -> StructureConstantAlgebra, Vector{AbsAlgAssMor}, Vector{AbsAlgAssMor}
direct_product(algebras::Vector{StructureConstantAlgebra}; task::Symbol = :sum)
  -> StructureConstantAlgebra, Vector{AbsAlgAssMor}, Vector{AbsAlgAssMor}

Returns the algebra $A=A_1\times \cdots \times A_k$. task can be ":sum", ":prod", ":both" or ":none" and determines which canonical maps are computed as well: ":sum" for the injections, ":prod" for the projections.

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direct_product(x::Vararg{T}) where T <: AbstractSpace -> T, Vector{AbstractSpaceMor}
direct_product(x::Vector{T}) where T <: AbstractSpace -> T, Vector{AbstractSpaceMor}

Given a collection of quadratic or hermitian spaces $V_1,\dots ,V_n$, return their direct product $V:=V_1\times \dots \times V_n$, together with the projections $V\to V_i$.

For objects of type AbstractSpace, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain V as a direct sum with the injections $V_i\to V$, one should call direct_sum(x). If one wants to obtain V as a biproduct with the injections $V_i\to V$ and the projections $V\to V_i$, one should call biproduct(x).

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# biproduct — Method.

biproduct(x::Vararg{T}) where T <: AbstractSpace -> T, Vector{AbstractSpaceMor}, Vector{AbstractSpaceMor}
biproduct(x::Vector{T}) where T <: AbstractSpace -> T, Vector{AbstractSpaceMor}, Vector{AbstractSpaceMor}

Given a collection of quadratic or hermitian spaces $V_1,\dots ,V_n$, return their biproduct $V:=V_1\oplus \dots \oplus V_n$, together with the injections $V_i\to V$ and the projections $V\to V_i$.

For objects of type AbstractSpace, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain V as a direct sum with the injections $V_i\to V$, one should call direct_sum(x). If one wants to obtain V as a direct product with the projections $V\to V_i$, one should call direct_product(x).

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Example

julia> E, b = cyclotomix_field_as_cm_extensions(7);
ERROR: UndefVarError: `cyclotomix_field_as_cm_extensions` not defined

julia> H = hermitian_space(E, 3);

julia> H2 = hermitian_space(E, E[-1 0 0; 0 1 0; 0 0 -1]);

julia> H3, inj, proj = biproduct(H, H2)
(Hermitian space of dimension 6, AbstractSpaceMor[Map: hermitian space -> hermitian space, Map: hermitian space -> hermitian space], AbstractSpaceMor[Map: hermitian space -> hermitian space, Map: hermitian space -> hermitian space])

julia> is_one(matrix(compose(inj[1], proj[1])))
true

julia> is_zero(matrix(compose(inj[1], proj[2])))
true

Orthogonality operations

# orthogonal_complement — Method.

orthogonal_complement(V::AbstractSpace, M::T) where T <: MatElem -> T

Given a space V and a subspace W with basis matrix M, return a basis matrix of the orthogonal complement of W inside V.

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# orthogonal_projection — Method.

orthogonal_projection(V::AbstractSpace, M::T) where T <: MatElem -> AbstractSpaceMor

Given a space V and a non-degenerate subspace W with basis matrix M, return the endomorphism of V corresponding to the projection onto the complement of W in V.

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Example

julia> K, a = cyclotomic_real_subfield(7);

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

julia> Q = quadratic_space(K, K[0 1; 1 0]);

julia> orthogonal_complement(Q, matrix(K, 1, 2, [1 0]))
[1   0]

---

Isotropic spaces

Let $(V,\mathrm{\Phi })$ be a space over $E/K$ and let $\mathfrak{p}$ be a place in $K$. $V$ is said to be isotropic locally at $\mathfrak{p}$ if there exists an element $x\in V_{\mathfrak{p}}$ such that ${\mathrm{\Phi }}_{\mathfrak{p}}(x,x)=0$, where ${\mathrm{\Phi }}_{\mathfrak{p}}$ is the continuous extension of $\mathrm{\Phi }$ to $V_{\mathfrak{p}}\times V_{\mathfrak{p}}$.

# is_isotropic — Method.

is_isotropic(V::AbstractSpace, p::Union{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, InfPlc}) -> Bool

Given a space V and a place p in the fixed field K of V, return whether the completion of V at p is isotropic.

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Example

julia> K, a = cyclotomic_real_subfield(7);

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

julia> E, b = number_field(t^2-a*t+1, "b");

julia> H = hermitian_space(E, 3);

julia> OK = maximal_order(K);

julia> p = prime_decomposition(OK, 7)[1][1];

julia> is_isotropic(H, p)
true

---

Hyperbolic spaces

Let $(V,\mathrm{\Phi })$ be a space over $E/K$ and let $\mathfrak{p}$ be a prime ideal of ${\mathcal{O}}_K$. $V$ is said to be hyperbolic locally at $\mathfrak{p}$ if the completion $V_{\mathfrak{p}}$ of $V$ can be decomposed as an orthogonal sum of hyperbolic planes. The hyperbolic plane is the space $(H,\mathrm{\Psi })$ of rank 2 over $E/K$ such that there exists a basis $e_1,e_2$ of $H$ such that $\mathrm{\Psi }(e_1,e_1)=\mathrm{\Psi }(e_2,e_2)=0$ and $\mathrm{\Psi }(e_1,e_2)=1$.

# is_locally_hyperbolic — Method.

is_locally_hyperbolic(V::Hermspace, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Bool

Return whether the completion of the hermitian space V over $E/K$ at the prime ideal p of ${\mathcal{O}}_K$ is hyperbolic.

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Example

julia> K, a = cyclotomic_real_subfield(7);

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

julia> E, b = number_field(t^2-a*t+1, "b");

julia> H = hermitian_space(E, 3);

julia> OK = maximal_order(K);

julia> p = prime_decomposition(OK, 7)[1][1];

julia> is_locally_hyperbolic(H, p)
false