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

Question

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.

Answer 1

Recall that finite fields must have order a power of a prime.

Moreover, if $k\subseteq k'$, then $k'$ is a vector space over $k$. Hence, if $|k|=p^n$, with $p$ prime, and $|k'|=p^m$, then if the dimension of $k'$ over $k$ is $d$ it follows that $p^m=(p^n)^d = p^{nd}$; that is, we must have that $n|m$.

In fact,

Theorem. Let $k$ and $k'$ be finite fields, with $|k|=p^n$ and $|k'|=q^m$, where $p$ and $q$ are primes and $n$ and $m$ are positive integers. Then $k\subseteq k'$ if and only if $p=q$ and $n|m$.

With this theorem, your problem turns into a problem about divisibility: for which integers $m$ is it the case that if $d_1$ and $d_2$ are divisors of $m$, then either $d_1|d_2$ or $d_2|d_1$?

(Or, if you go via the Galois correspondence, as suggested by Mariano, you are asking: which finite cyclic groups have the property that their subgroups form a chain under inclusion?)

Answer 2

First read Mariano's comment and Arturo's answer. If that is not enough, then...

Take a look at the diagram in this question. Ignore that extraneous long vertical line connecting the field at the bottom with the one at the top. You notice that the subfields of $\mathbb{F}_{2^{12}}$ don't form a chain, as required in your exercise. For example, neither of the intermediate fields $\mathbb{F}_4$ and $\mathbb{F}_8$ is a subfield of the other. But there is clue in that diagram that allows us to diagnose, why that happened!

1. In that diagram two subfields are connected with a line, iff one is an extension of the other, but there are no intermediate fields between them. Why don't you begin by writing the degree of the extension (of the two fields involved) adjacent to each line? Notice something about those numbers?

The numbers are prime factors of 12. In general you will need to use $[k:\mathbb{F}_p]$ in place of 12.

2. Look at those fields in the diagram, where "branching" occurs, i.e. those fields that are at the bottom of two (or in a more general case two or more) lines. Notice something about the numbers attached to those lines?

The prime numbers on those lines are distinct.

3. How could we prevent that from ever happening?

If $p$ and $q$ are two different prime factors of $n$, and $G$ is a cyclic group of order $n$ generated by $g$, then $g^p$ and $g^q$ generate subgroups of orders $n/p$ and $n/q$ respectively. What does Galois theory tell you about the fixed fields of those two subgroups?