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The task is the following:

Show that a Galois extension $L/K$ of degree $45$ has got at most $12$ intermediate field extensions.

Below I present a proof. I seek a more general method for this kind of problems.

Using Galois correspondence, each intermediate field extension corresponds to a subgroup of $Gal(L/K)$ and $|Gal(L/K)|=45$.

Using Sylow theorems, we show that there are one of each Sylow subgroups for $3,5$ and conclude that they are normal subgroups and thus their product generates the whole group.

The Sylow-$5$ group must be isomorphic to $\Bbb{Z}_5$ and the Sylow-$3$ group must be isomorphic to either $\Bbb{Z}_9$ or $\Bbb{Z}_3 \times \Bbb{Z}_3$.

$\Bbb{Z}_5 \times \Bbb{Z}_9 \cong\Bbb{Z}_{45}$ is cyclic so every divisor of $45$ gives exactly one subgroup, total of $6$.

$\Bbb{Z}_5 \times \Bbb{Z}_3 \times \Bbb{Z}_3$ has the subgroups:

  • Order $1$: $\langle 1 \rangle$
  • Order $3$: $\langle (010) \rangle$, $\langle (001) \rangle$, $\langle (011) \rangle$, $\langle (012) \rangle$
  • Order $5$: $\langle (100) \rangle$
  • Order $9$: $\langle (010),(001) \rangle$
  • Order $15$: $\langle (100), \sigma \rangle$ for each $\sigma$ of order $3$, total of $4$
  • Order $45$: whole group

Which gives exactly $12$ subgroups.

However the last part seemed like a proof by exhaustion and it seemed "lucky", the group was pretty small and I could just write out the subgroups.

Are there more elegant methods for this problem? General ones or in the case that the group is a product of normal subgroups? Thanks in advance.

B.Swan
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    By :"questions of this form", do you mean questions of the form : "show that a Galois extension of size $x$ has at most $y$ intermediate fields"? This then boils down to the question : "show that any group of size $x$ has at most $y$ subgroups". Now, do you want answers to this question which is group theoretic? Or by generalization do you want techniques to do this for even non-Galois finite extensions, for example? – Sarvesh Ravichandran Iyer Mar 18 '19 at 16:27
  • I would welcome a group theoretic answer, but I would like the answer to be as general as possible. If it is possible with less assumptions about the field extension, I would like to know what one can do then. – B.Swan Mar 18 '19 at 18:15
  • There are inseparable field extensions $L/K$ of degree $45$ with infinitely many intermediate field extensions. So omitting separability entirely doesn't lead to any decent generalization. So in which direction would you like to generalize the problem? – Servaes Mar 19 '19 at 10:29
  • I would like to know when the number of intermediate fields is limited and what the methods of showing it then are. If its the case for Galois extensions, then let us only consider those. If there are some special non-Galois cases, you can mention those if they are worth including. – B.Swan Mar 19 '19 at 12:43

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