Exercise 3, § 16

Let E be an extension of a field K and F an extension of E. a) If E admits a separating transcendence basis over K and if F admits a separating transcendence basis over E, then F admits a separating transcendence basis over K. b) If F admits a separating transcendence basis over K and if $\operatorname{tr. deg}_K E < +\infty$, then E admits a separating transcendence basis over K (reduce to the case where $\operatorname{tr. deg}_K F < +\infty$ and apply Ex. 1). c) If F admits a finite separating transcendence basis over K and is separable over E then F admits a separating transcendence basis over E (use Ex. 1).