Arason–Pfister Hauptsatz

Remarks

The bound is sharp: an anisotropic $n$-fold Pfister form lies in $I^n(F)$ and has dimension exactly $2^n$. Moreover an anisotropic form in $I^n(F)$ of dimension $2^n$ is similar to an $n$-fold Pfister form. The proof by Arason and Pfister uses Pfister forms and their behaviour over function fields of quadrics. Example: for $n = 1$, a nonzero anisotropic form of even dimension has dimension at least $2$, which is clear. For $n = 2$, a $2$-dimensional form in $I^2(F)$ has trivial discriminant, so it is $\langle a, -a \rangle$, which is hyperbolic; hence anisotropic forms in $I^2(F)$ have dimension at least $4$. A consequence is that $\bigcap_{n} I^n(F) = 0$, since each nonzero Witt class contains an anisotropic form of some fixed dimension.

Going deeper

For a student

Hauptsatz is German for main theorem. The theorem gives the separation property $\bigcap_n I^n(F) = 0$, which is needed to use the graded ring $\bigoplus_n I^n(F)/I^{n+1}(F)$ to study the Witt ring. Together with the Milnor conjecture on quadratic forms, it means that a quadratic form is determined up to Witt equivalence by invariants with values in Galois cohomology.