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Find all finite fields $k$ whose subfields form a chain: that is, if $k'$ and $k''$ are subfields of $k$, then either $k' \subseteq k''$ or $k'' \subseteq k'$.

So, I understand that I'm trying to find the values of $n$ such that the subfields of $\mathbb{F}_{p^n}$ (where $\mathbb{F_{p^n}}$ is the Galois field of order $p^n$) form a chain. However, I'm unsure of where to begin. If anyone could point me in the right direction that would really help. This problem is found in Rotman's Advanced Modern Algebra.

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    This will happen iff the subgroups of the Galois group of the extension has its subgroups totally ordered under inclusion. Does this help?2012-05-15

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