Dear Marc, Erik, Forum,
I have a finitely generated matrix algebra over the rationals that I know is
not a field but contains
zero divisors.
To find one (and this seems to be quite suitable) I simply use:
i:=First(rc.algebra, a-> Determinant(a)=0 and a <> 0*I);
Does anybody know how to do this another way with more chance for a success?
In general this is a very difficult problem. If the input basis is
bad, then no general algorithm is known (if the algebra is central simple).
For more information:
@article {MR2325916,
AUTHOR = {P{\'{\i}}lnikov{\'a}, Jana},
TITLE = {Trivializing a central simple algebra of degree 4 over the
rational numbers},
JOURNAL = {J. Symbolic Comput.},
FJOURNAL = {Journal of Symbolic Computation},
VOLUME = {42},
YEAR = {2007},
NUMBER = {6},
PAGES = {579--586},
ISSN = {0747-7171},
MRCLASS = {16K20},
MRNUMBER = {2325916 (2008c:16030)},
MRREVIEWER = {Zinovy Reichstein},
DOI = {10.1016/j.jsc.2007.01.001},
URL = {http://dx.doi.org/10.1016/j.jsc.2007.01.001},
}
Best wishes,
Willem de Graaf
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