Quaternion algebras

We provide a model for quaternion algebras over a field $K$ in standard form, which is parametrized by two elements $a,b\in K$. The corresponding quaternion algebra has a basis $1,i,j,k$ satisfying $i^2=a$, $j^2=b$ and $ij=-ji=k$.

This functionality is currently restricted to fields of characteristic not equal to two.

Creation

quaternion_algebra Method

quaternion_algebra(K::Field, a, b) -> QuaternionAlgebra

If the characteristic of K isn't 2, return the quaternion algebra $(a,b{)}_K$ defined by $i^2=a$, $j^2=b$, and $ij=-ji$.

If the characteristic of K is 2, return the quaternion algebra $[a,b{)}_K$ defined by $i^2+i=a$, $j^2=b$, and $ij=ji+j$.

Examples

julia> Q = quaternion_algebra(QQ, -1, -1)
Quaternion algebra
  over rational field
  defined by i^2 = -1, j^2 = -1, ij = -ji

julia> K, sqrt2 = quadratic_field(2);

julia> Q = quaternion_algebra(K, sqrt2, -1)
Quaternion algebra
  over real quadratic field defined by x^2 - 2
  defined by i^2 = sqrt(2), j^2 = -1, ij = -ji

julia> R, x = polynomial_ring(GF(2), "x"); F=fraction_field(R);

julia> Q = quaternion_algebra(F, 0, x)
Quaternion algebra
  over fraction field of R
  defined by i^2 = i+0, j^2 = x, ij = ji+j

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Arithmetic of elements

conjugate Method

conjugate(a::AssociativeAlgebraElem{_, QuaternionAlgebra})
                             -> AssociativeAlgebraElem{_, QuaternionAlgebra}

Return the image of $a$ under the canonical involution of the quaternion algebra.

Examples

julia> Q = quaternion_algebra(QQ, -1, -1); a = Q([1, 1, 1, 1])
1 + i + j + k

julia> conjugate(a)
1 - i - j - k

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trred Method

trred(x::AbstractAssociativeAlgebraElem{T}) where T -> T

Returns the reduced trace of $x$.

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normred Method

normred(x::AbstractAssociativeAlgebraElem{T}) where T -> T

Returns the reduced norm of $x$.

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reduced_charpoly Method

reduced_charpoly(a::AbstractAssociativeAlgebraElem) -> PolyRingElem

Returns the reduced characteristic polynomial of $a$ as a polynomial over base_ring(algebra(a)).

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Example

julia> Q = quaternion_algebra(QQ, -1, -1)
Quaternion algebra
  over rational field
  defined by i^2 = -1, j^2 = -1, ij = -ji

julia> z = Q([1, 2, 0, 1//3])
1 + 2*i + 1//3*k

julia> trred(z)
2

julia> normred(z)
46//9

julia> reduced_charpoly(z)
x^2 - 2*x + 46//9

julia> m = reduced_charpoly(z)
x^2 - 2*x + 46//9

julia> m(z)
0

Splitting of quaternion algebras

is_split_with_zero_divisor Function

is_split_with_zero_divisor(A::QuaternionAlgebra) -> Bool, AssociativeAlgebraElem

Given a quaternion algebra $A$, return whether $A$ is split together with an element, which is a zero-divisor in case $A$ is split.

Examples

julia> A = quaternion_algebra(QQ, 1, 4);

julia> is_split_with_zero_divisor(A)
(true, 1 + i)

julia> A = quaternion_algebra(QQ, -1, -1);

julia> is_split_with_zero_divisor(A)
(false, 0)

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