Let be a cyclic extension of dimension over and let be a generator of . Let , and suppose is a non-zero element of such that for some . Show that there exists a in the (unique) subfield of of dimensionality such that .

is a cyclic group, then it has a unique subgroup of index . By the Galois corresponding theorem, there exists a unique subfield such that . . has dimensionality over .

Consider , then . We also have

,

by Hilbert’s Satz 90, such that .

Let , then , because

Surprisingly we have

.

Jacabson p300. This problem puzzled me for three weeks. Finally it turns out to be very easy.