Suppose such an extension exists, whence we have a tower of field extensions , so in particular divides , i.e. .

Conversely, assume , whence , and by the same token . Therefore . By Construction and uniqueness, is the splitting field of the second polynomial. It follows from ^P1 that .

For the last statement, note that Finite subgroup of the group of units of a field is cyclic, so in particular has a generator , which will necessarily generate over any subfield. If this means , so the extension is simple.