next up previous contents
Next: uniqueness of an operator Up: First properties of Weyl Previous: Algebra endomorphism and splitting   Contents

the geometric degree

Definition 1   Let $ k$ be a field of positive characteristic. Then for any $ k$-algebra homomorphism. $ \phi:\overline{A_n(k)}\to A_n(k)$, the geometric degree $ \operatorname{geomdeg}(\phi)$ of $ \phi$ is defined to be the index $ [Q(Z_n(k)):Q(\phi(Z_n(k))]$ of the corresponding field extension.

Note that if the geometric degree is finite, then by comparing the transcendent degree we see that $ \phi$ is actually injective and the geometric degree is equal to $ [K_n(k):\overline{K_n(k)}]$.



2003/3/3