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Let $R$ be an integral domain over a field $k$. Is it true, that $\deg.\mathrm{tr}_k \ \mathrm{Frac}(R)$ is the greatest number of elements of $R$ algebraically independent over $k$?

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    Yes: http://mathoverflow.net/questions/752$1$92012-10-28

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Every transcendence basis of an algebra is a transcendence basis of its quotient field. Because the set of algebraic elements over subfield is a subfield, and the lowest subfield containing algebra is its quotient field.

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    It means, that they are algebraically independent over $k$ and for every $r \in R$ the elements $u_1, ..., u_n$ are dependent.2012-10-29