Let
Consider
Then Monsky proved the following results on the Hilbert-Kunz multiplicity of the maximal ideal ( x , y , z ) {\displaystyle {}(x,y,z)} in S ⊗ F 2 [ t ] L {\displaystyle {}S\otimes _{{\mathbb {F} }_{2}[t]}L} , L {\displaystyle {}L} a field:
![{\displaystyle {}S={\mathbb {F} }_{2}[t,x,y,z]/(g)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/646f9b959d94309c075d88b8fe0d9171a5ed5157)
in ![{\displaystyle {}S\otimes _{{\mathbb {F} }_{2}[t]}L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b301fe3296177d33c86f033a751465e422ef6e50)
, 
a field:
![{\displaystyle {}e_{HK}(S\otimes _{\mathbb {F} _{2}[t]}L)={\begin{cases}3{\text{ for }}L={\mathbb {F} }_{2}(t)\\3+{\frac {1}{4^{d}}}{\text{ for }}L={\mathbb {F} }_{q}={\mathbb {F} }_{2}(\alpha ),\,(t\mapsto \alpha ,\,d=\deg(\alpha ))\,.\end{cases}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5164f2995bad1fdb05228b6476ca55e65806cf39)
