Trdeg

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In mathematics, usually algebra, trdeg is the degree of transcendentality between two fields, more precisely, consider two fields F and E, the \operatorname{trdeg}(E/F) is defined to be the transcendentality of E over F.

By considering trdeg we can get an intuitive feeling of how large a field can be. For instance, in sheaf theory a theorem due to Siegel states that

Let X be a compact connected complex manifold of dimension n. Then

\text{trdeg}(K(X)/\mathbb{C})\le n

where K(X) denotes the space of global sections of the sheaf of meromorphic functions.

By this theorem, we learn that the transcendentality of global meromorphic function is at most the dimension of the underlying complex manifold.

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