I’d like to get some help about an exercise in Field theory (J. Bastida, ‘Field extensions and Galois theory’, p. 157, problem 2).
Let K be a field and let L be a separable extension of K. Suppose that there exists a positive integer n such that [K(α):K] is less of equal than n for every α in L. Prove that L is finite over K and [L:K] is less of equal than n.
1 proves it in characteristic 0, but I’d like a proof independent of the characteristic. There they say the result is proved in Lang’s ‘Undergraduate Algebra’, but I haven’t found it yet. A hint (found in Bosch, ‘Algebra’) is to apply the primitive element theorem (which he states as: every finite separable extension is simple) to the subfields of L containing K and of finite degree over it.