The Stacks project

Example 9.26.6. Consider the field extension $\mathbf{Q}(e, \pi )$ formed by adjoining the numbers $e$ and $\pi $. This field extension has transcendence degree at least $1$ since both $e$ and $\pi $ are transcendental over the rationals. However, this field extension might have transcendence degree $2$ if $e$ and $\pi $ are algebraically independent. Whether or not this is true is unknown and whence the problem of determining $\text{trdeg}(\mathbf{Q}(e, \pi ))$ is open.