I have to show no proper intermediate fields exist between $Z_2$ and $GF(2^3)$

Question

I have to show no proper intermediate fields exist between $Z_2$ and its overfield $GF(2^3)$, Can any one help?

Have you tried computing the degree of the extension? – Pierre-Guy Plamondon Jul 29 '15 at 10:59 — Pierre-Guy Plamondon, Jul 29 '15 at 10:59

score 0 · Answer 1 · answered Jul 29 '15 at 11:56

0

It is a theorem on finite fields that for any prime $p$, and any positive natural numbers, $$\mathbf F_{p^{\scriptstyle r}}\subset \mathbf F_{p^{\scriptstyle s}}\iff r\mid s.$$

answered Jul 29 '15 at 11:56

Bernard

179,256

I have to show no proper intermediate fields exist between $Z_2$ and $GF(2^3)$

1 Answers1