Question: Given a semisimple finite dimensional algebra over the rationals in GAP. Is there a way to find a (minimal) splitting field of this algebra with GAP and an explicit Wedderburn decomposition?
The semisimple algebra is given as a subalgebra of the ring of $n \times n$-matrices of the matrix algebra over the rationals. It is not a group algebra in general.
The GAP-command DirectSumDecomposition seems to give a decomposition of such an algebra into two-sided ideals (so a rough Wedderburn decomposition). I wonder whether there is a more detailed way to see what those two-sided ideals $L$ are exactly meaning to know $L \cong M_n(D)$ for $n$ a number and $D$ a division ring over the rationals.