On the first Galois cohomology group of the algebraic group SL1(D)

Let A be a central simple algebra over a field F. Denote the reduced norm of A over F by Nrd: A* → F* and its kernel by SL1(A). For a field extension K of F, we study the first Galois Cohomology group H 1(K,SL1(A)) by two methods, valuation theory for division algebras and K-theory. We shall show that this group fails to be stable under purely transcendental extension and formal Laurent series.