Transcendence degree for a $k$-algebra which is an integral domain 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 $k$?
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.
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.