Answer to Question #23570 in Abstract Algebra for jeremy

Question

If K ⊇ k is a splitting field for a finite-dimensional k-algebra R, does it follow that K is also a splitting field for any quotient algebra R of R?

Answer 1

Since this fails for subalgebras, animpulsive guess might be that it also fails for quotient algebras. However, the
statement turns out to be true for quotient algebras. To prove this, we may
assume that K = k. Consider the natural surjection ϕ : R → R'. Any simple left R'-moduleV may be viewed as a simple left R-module via ϕ. For any field extension L ⊇ k, V ^L remains simple as aleft R^L-module. Therefore, V ^L is also asimple left R'^L-module. This shows that V isabsolutely irreducible, so k is indeed a splitting field for R.