Remark 3

Suppose that the field K satisfies the following condition:

– If L is an extension of K of finite degree, then it is cyclic and the norm mapping $N : L^*\rightarrow K^*$ is surjective.

This condition is satisfied, in particular, when the field K is finite (V, §12, No. 2, p. 95, Proposition 4). We can then prove that every field of finite degree over K with center K is equal to K (Exercise 10 of VIII, p. 329).