Module: sage.algebras.quaternion_algebra
Author: David Kohel, 2005-09
Module-level Functions
K, a, b, [denom=None], [names=1]) |
Return the quaternion algebra over
generated by
,
, and
such that
,
, and
.
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]
D1, D2, T, [M=None], [names=2]) |
Return the quaternion algebra over the rationals generated by
,
, and
where
,
, and
are
quadratic suborders of discriminants
,
, and
, respectively. The traces of
and
are
chosen in
.
The integers
,
and
must all be even or all odd, and
,
and
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
K, gram, [names=None]) |
K, norms, traces, [names=None]) |
D) |
a, b) |
D1, D2, T) |
x) |
Class: QuaternionAlgebra_generic
self, K, [ramified_primes=None]) |
Functions: basis,
discriminant,
gen,
gram_matrix,
inner_product_matrix,
ramified_primes,
random_element,
vector_space
self, i) |
The i-th generator of the quaternion algebra.
Special Functions: __call__,
__repr__,
_set_ramified_primes