Vertex operator algebras associated to the Virasoro algebra over an arbitrary field

The vertex operator algebras and modules associated to the highest weight modules for the Virasoro algebra over an arbitrary field $\mathbb {F}$ with $\mathrm {ch}\mathbb {F}\ne 2$ are studied. The irreducible modules of vertex operator algebra $L(\frac {1}{2},0)_{\mathbb {F}}$ are classified. The rationality of $L(\frac {1}{2},0)_{\mathbb {F}}$ is established if $\mathrm {ch}\mathbb {F}\ne 7.$