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A quaternion algebra A over a field K is a central simple algebra
of dimension four over K, or equivalently when K does not have
characteristic 2, an algebra generated by elements i, j which satisfy
i2=a, j2=b, ji= - ij
with a, b ∈K * . A quaternion algebra in Magma is a specialized
type AlgQuat, which is a subtype of associative algebra AlgAss
defined over a field. Magma can recognize if an associative algebra
A is a quaternion algebra and will return a standard representation
for A with a, b as above.
Examples of quaternion algebras include the ring M2(K) of
2 x 2-matrices over K with a=b=1, as well as the division
ring of Hamiltonians over K=R with a=b= - 1. Every
quaternion algebra over a finite field or an algebraically closed
field is isomorphic to M2(K). Over a local field, there is a unique
quaternion algebra (up to isomorphism) which is a division ring.
Every quaternion algebra A over K not isomorphic to M2(K) is a
division algebra. Finding a zerodivisor in A is computationally
equivalent to the problem of finding a K-rational point on a conic.
Given a zerodivisor in A, Magma will compute an
explicit isomorphism A to M2(K). If L is an extension field
of K, then AL=A tensor K L is a quaternion algebra over L,
and we say L is a splitting field if AL isomorphic to M2(L). If
[L:K]=2, then L is a splitting field if and only if there exists
a K-embedding L -> A, and such an embedding can
be computed in Magma.
Further functionality is available for quaternion algebras A defined
over number fields K. Such an algebra is said to be unramified
at a noncomplex place v of K if Kv is a splitting field for A,
otherwise A is ramified at v.
Testing if an algebra is ramified at a place is encoded in the
Hilbert symbol, which can be computed for any such algebra.
The set S of ramified places of an algebra is finite and of even
cardinality, and conversely, given such a set S there exists a
quaternion algebra which is ramified only at S. One may
compute the set of ramified places of A as well
as constructing an algebra given its ramification set.
In Magma, there are algorithms for quaternion algebras
which are analogous to those for number fields---computation
of the ring of integers, class group,
and unit group. One may compute a maximal order
O ⊂A, a p-maximal order for a prime p of K, and orders
with given index in a maximal order. Secondly, a representative
set of (right) ideal classes in O can be enumerated, and one can
test if two ideals are isomorphic (hence if a given
ideal is principal).
Finally, for a definite quaternion algebra (defined over a totally
real field), one may compute the group of units of norm 1 in O.
Over Q, these functions tie into machinery for constructing
the Brandt module, with applications to modular forms.
Orders for quaternion algebras over the rational numbers and over
rational function fields in Magma
have type AlgQuatOrd and ideals have type AlgQuatOrdIdl.
Over other number rings, orders have the general type AlgAssVOrd
and ideals AlgAssVOrdIdl which is also the type for used
for associative orders, see Section Orders for information about these
orders. The types AlgQuatOrd,
AlgQuatOrdElt and AlgQuatOrdIdl
inherit from the types AlgAssVOrd, AlgAssVOrdElt and
AlgAssVOrdIdl respectively.
The main reference for material in this chapter is the book of
Vignéras [Vig80]. Most nontrivial algorithms for quaternion
algebras over the rationals and over number fields are described in
[KV10].
IMPORTANT WARNING.
In Magma, the rationals are not considered to be a number field
(the type FldRat is not a subtype of FldNum).
Much of the functionality for quaternion algebras, and particularly
for orders in quaternion algebras, is implemented only for algebras
whose base field is a FldNum (while some, but not all, also works
for algebras over the FldRat). To compute with algebras over
Q, in many cases the best solution is to create Q as a number field
at the outset, using RationalsAsNumberField(), and create the algebra
over this field instead of Rationals().
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