#### Vol. 4, No. 8, 2010

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A lower bound on the essential dimension of simple algebras

### Alexander S. Merkurjev

Vol. 4 (2010), No. 8, 1055–1076
##### Abstract

Let $p$ be a prime integer and $F$ a field of characteristic different from $p$. We prove that the essential $p$-dimension ed${}_{p}\left(\rightCSA\left({p}^{r}\right)\left)\right$ of the class $CSA\left({p}^{r}\right)$ of central simple algebras of degree ${p}^{r}$ is at least $\left(r-1\right){p}^{r}+1$. The integer ${ed}_{p}\left(\rightCSA\left({p}^{r}\right)\left)\right$ measures complexity of the class of central simple algebras of degree ${p}^{r}$ over field extensions of $F$.

##### Keywords
essential dimension, Brauer group, algebraic tori
##### Mathematical Subject Classification 2000
Primary: 16K50
Secondary: 14L30, 20G15