17.1 Quaternion algebras

Module: sage.algebras.quaternion_algebra

Author: David Kohel, 2005-09

Module-level Functions

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
sage: A.ramified_primes()
[2]

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])

fundamental_discriminant( D)

ramified_primes( a, b)

ramified_primes_from_discs( D1, D2, T)

sign( x)

Class: QuaternionAlgebra_generic

class QuaternionAlgebra_generic
QuaternionAlgebra_generic( self, K, [ramified_primes=None])

Functions: basis,$  $ discriminant,$  $ gen,$  $ gram_matrix,$  $ inner_product_matrix,$  $ ramified_primes,$  $ random_element,$  $ vector_space

gen( self, i)

The i-th generator of the quaternion algebra.

Special Functions: __call__,$  $ __repr__,$  $ _set_ramified_primes

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