#### Z. Reichstein, * On the Notion of Essential Dimension
for Algebraic Groups *, Transformation Groups, 5,
200 (2001), 207-249.

** Abstract: **
We introduce and study the notion of essential
dimension for linear algebraic groups defined over an algebraically
closed fields of characteristic zero.
The essential dimension is a numerical invariant of the group;
it is often equal to the minimal number of independent
parameters required to describe all algebraic objects
of a certain type. For example, if our group G is
S_n, these objects are field extensions, if
G = O_n, they are quadratic forms, if G = PGL_n, they are
division algebras (all of degree n),
if G = G_2, they are octonion algebras,
if G = F_4, they are exceptional Jordan algebras.
We develop a general theory, then compute or estimate
the essential dimension for a number of specific groups, including
all of the above-mentioned examples. In the last section we give
an exposition of results, communicated to us by J.-P. Serre, which
relate the essential dimension of G to cohomological invariants
of principal G-bundles.

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