Exercise 13, § 11

A field K is called a Moriya field if every separable algebraic extension of finite degree over K is cyclic. \* Every finite field is a Moriya field (V, p. 95, Prop. 4). \* The field M of Ex. 12 is a Moriya field. a) For K to be a Moriya field it is necessary and sufficient that for each integer $n > 0$ there should exist at most one separable extension of degree $n$ over K. (Use the following fact: if G is a finite group and for each integer $m \geq 1$ dividing the order of G there exists at most one subgroup of G of order m, then G is cyclic; this follows from the same property for p-groups (Ex. 12) and from I, p. 80, Th. 4). b) If K is a Moriya field, show that every algebraic extension E of K is a Moriya field and that if F is an extension of E of degree $m$ over E, then there exists an extension of K of degree m over K.