Degree of vector bundle under extension of scalars.

Question

I have a somewhat of a trivial question, just want to confirm it (or maybe its wrong). Degree of a vector bundle doesn't change when we extend the scalars is that right? So if a vector bundle is semi-stable after extension to algebraic closure, it had to be semi-stable from the beginning, or is there a counter-example?

Answer 1

The degree of a bundle, of course, does not change under extensions of scalars and semi-stability is preserved, because the Harder--Narasimhan filtration is canonical, hence Galois-invariant, hence descends from the algebraic closure of the base field.