Essential dimension

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,...,e_n of V such that q can be expressed in the form ${\displaystyle q(\sum x_{i}e_{i})=\sum a_{ij}x_{i}x_{j}}$ with all coefficients a_ij 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.

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