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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. ==Formal definition== Fix an arbitrary field k and let ''Fields/k'' denote the category of finitely generated field extensions of k with inclusions as morphisms. Consider a (covariant) functor F : Fields/k → Set. For a field extension K/k and an element ''a'' of F(K/k) a ''field of definition of a'' is an intermediate field K/L/k such that ''a'' is contained in the image of the map F(L/k) → F(K/k) induced by the inclusion of L in K. The ''essential dimension of a'', denoted by ''ed(a)'', is the least transcendence degree (over k) of a field of definition for ''a''. The essential dimension of the functor F, denoted by ''ed(F)'', is the supremum of ''ed(a)'' taken over all elements ''a'' of F(K/k) and objects K/k of Fields/k. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Essential dimension」の詳細全文を読む スポンサード リンク
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