12. The Essential Dimension of Central Simple Algebras

Roughly speaking, the essential dimension of an algebraic structure measures the complexity of the structure by giving the number of independent parameters needed to define it. For a prime number p, the essential p-dimension is a variation of essential dimension that takes into account simplifications in structure that can occur after field extensions of degree prime to p. In §12.1 we define the basic notions of essential dimension and essential p-dimension. We also give examples, which emphasize the relation with the decomposability of central simple algebras into tensor products of symbols. Merkurjev in [150] and Baek–Merkurjev in [21] recognized that the presence of a valuation on a division algebra D implies some level of complexity to D, and they used this to give a lower bound on the essential p-dimension of D. In §12.2 we give a version of the valuation-theoretic part of their arguments. Then in §12.3 we sketch their method to obtain lower bounds on the essential p-dimension of central simple algebras. In the final §12.4, we discuss consequences of these lower bounds for the decomposability of division algebras into tensor products of symbol algebras.