Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 21 / 1 / 5 / Solution

Put $d=\mathrm{deg}D$, choose a $K$-basis of $D$, and let $N(X_1,\dots ,X_{d^2})$ be its reduced norm polynomial. For $a\in K^{\times }$, considerThis has degree $d$ and $d^2+1>d^2$ variables. The $C_2$ property gives a nontrivial zero $(x,t)$. Necessarily $t\ne 0$: otherwise $N(x)=0$ forces $x=0$ in the division algebra, making the whole coordinate vector zero. Homogeneity now gives $N(x/t)=a$. The value zero is achieved by the zero element. HenceThis is the surjectivity of reduced norms over C2 fields, and the proof also gives the multiplicative surjection $D^{\times }\to K^{\times }$. The argument works for degree one as well.