In mathematics, essential dimension is an invariant defined for certain algebraic structures such as algebraic groups and quadratic forms. It was introduced by J. Buhler and Z. Reichstein and in its most generality defined by A. Merkurjev.
Basically, essential dimension measures the complexity of algebraic structures via their fields of definition. For example, a quadratic form q : V → K over a field K, where V is a K-vector space, is said to be defined over a subfield L of K if there exists a K-basis e1,...,en of V such that q can be expressed in the form with all coefficients aij belonging to L. If K has characteristic different from 2, every quadratic form is diagonalizable. Therefore q has a field of definition generated by n elements. Technically, one always works over a (fixed) base field k and the fields K and L in consideration are supposed to contain k. The essential dimension of q is then defined as the least transcendence degree over k of a subfield L of K over which q is defined.
Read more about Essential Dimension: Formal Definition, Examples, Known Results
Famous quotes containing the words essential and/or dimension:
“Those of us who are in this world to educate—to care for—young children have a special calling: a calling that has very little to do with the collection of expensive possessions but has a lot to do with the worth inside of heads and hearts. In fact, that’s our domain: the heads and hearts of the next generation, the thoughts and feelings of the future.”
—Fred M. Rogers, U.S. writer and host of Mr. Rogers Neighborhood. “That Which is Essential Is Invisible to the Eye,” Young Children (July 1994)
“Le Corbusier was the sort of relentlessly rational intellectual that only France loves wholeheartedly, the logician who flies higher and higher in ever-decreasing circles until, with one last, utterly inevitable induction, he disappears up his own fundamental aperture and emerges in the fourth dimension as a needle-thin umber bird.”
—Tom Wolfe (b. 1931)