The norm theorem for semisingular quadratic forms

Let F be a field of characteristic 2. The aim of this paper is to give a complete proof of the norm theorem for singular F-quadratic forms which are not totally singular, i.e., we give necessary and sufficient conditions for which a normed irreducible polynomial of $F[x_1,\dots ,x_n]$ becomes a norm of such a quadratic form over the rational function field $F(x_1,\dots ,x_n)$. This completes partial results proved on this question in [8]. Combining the present work with the papers [1] and [7], we obtain the norm theorem for any type of quadratic forms in characteristic 2.