## Some consequences of the Karpenko-Merkurjev theorem.

### Summary

We use a recent theorem of N. A. Karpenko and A. S. Merkurjev to settle several questions in the theory of essential dimension.

### Mathematics Subject Classification

20G15, 20D15, 20C15, 14L30, 11E72

### Keywords/Phrases

essential dimension, linear representations, finite $p$-groups, algebraic tori