Proposition 6

For every integer $r$ such that $0\leqslant r\leqslant n$, the mapping $b_r$ from A to K is a homogeneous polynomial mapping of degree $r$. In particular, the reduced norm is a homogeneous polynomial mapping of degree $n$ from A to K.

Let $(e_i)_{i\in I}$ be a basis of A over K and $\mathbf{T}= (T_i)_{i\in I}$ a family of variables.