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.
Read more about this topic: Essential Dimension
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