9.3.1 algebras.quaternion_algebra - Quaternion algebras

AUTHOR: David Kohel, 2005-09

The module algebras.quaternion_algebra defines the following methods:

QuaternionAlgebra( K, a, b, [denom=None], [names=1])

Return the quaternion algebra over $ K$ generated by $ i$ , $ j$ , and $ k$ such that $ i^2 = a$ , $ j^2 = b$ , and $ ij=-ji=k$ .

INPUT:
    K -- field
    a -- element of K
    b -- element of K
    denom -- (optional, default 1)
    names -- list of three strings

sage: A = QuaternionAlgebra(QQ, -1,-1, names=list('ijk'))
sage: i,j,k = A.gens()
sage: i**2
-1
sage: j**2
-1
sage: i*j
k
sage: j*i
-k
sage: (i+j+k)**2
-3

QuaternionAlgebraWithDiscriminants( D1, D2, T, [M=None], [names=2])

Return the quaternion algebra over the rationals generated by $ i$ , $ j$ , and $ k = (ij - ji)/M$ where $ \mathbf{Z}[i]$ , $ \mathbf{Z}[j]$ , and $ \mathbf{Z}[k]$ are quadratic suborders of discriminants $ D_1$ , $ D_2$ , and $ D_3 = (D_1
D_2 - T^2)/M^2$ , respectively. The traces of $ i$ and $ j$ are chosen in $ \{0,1\}$ .

The integers $ D_1$ , $ D_2$ and $ T$ must all be even or all odd, and $ D_1$ , $ D_2$ and $ D_3$ must each be the discriminant of some quadratic order, i.e. nonsquare integers = 0, 1 (mod 4).

INPUT:
    D1 -- Integer
    D2 -- Integer
    T  -- Integer
    
OUTPUT:
    A quaternion algebra.

sage: A = QuaternionAlgebraWithDiscriminants(-7,-47,1, names=['i','j','k'])
sage: print A
Quaternion algebra with generators (i, j, k) over Rational Field
sage: i, j, k = A.gens()
sage: i**2
-2 + i
sage: j**2
-12 + j
sage: k**2
-24 + k
sage: i.minimal_polynomial()
x^2 - x + 2
sage: j.minimal_polynomial()
x^2 - x + 12

QuaternionAlgebraWithGramMatrix( K, gram, [names=None])

QuaternionAlgebraWithInnerProduct( K, norms, traces, [names=None])

hilbert_symbol( a, b, p, [algorithm=None])

Returns 1 if $ ax^2 + by^2$ $ p$ -adically represents a nonzero square, otherwise returns $ -1$ .

The module algebras.quaternion_algebra defines the following classes:

class QuaternionAlgebra_generic



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