Timeline for How to distinguish division algebras from matrix algebras?
Current License: CC BY-SA 2.5
3 events
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| Mar 6, 2011 at 11:13 | comment | added | Mikhail Bondarko | Actually, trace is not really relevant. You just need any $k$-linear map $f:K\to k$ such that $f(x^{-1})\neq 0$. Then we have $id_K\otimes f(x\otimes x^{-1})=x\otimes ff(x^{-1})$; so you can recover $x$ (up to a $k$-multiplier that does not matter anyway). Probably the method that associates a 2-cocycle to $D$ could be found in the original paper of Amitsur: ams.org/journals/tran/1959-090-01/S0002-9947-1959-0101265-7/… | |
| Mar 6, 2011 at 8:44 | comment | added | Tim Dokchitser | Thank you! This approach is new to me, so could I perhaps ask you to add a few more details? I can certainly find a (non-Galois) splitting field $K$ by taking a subfield of $D$; how do I can compute the class of $D$ in $(K\otimes K)^\times$ and how does this trace map help exactly? | |
| Mar 5, 2011 at 23:16 | history | answered | Mikhail Bondarko | CC BY-SA 2.5 |