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^[1] and in its most generality defined by A. Merkurjev.^[2]

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 ${\displaystyle q\left(\sum x_{i}e_{i}\right)=\sum a_{ij}x_{i}x_{j}}$ 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.