Subalgebra lattices of finite fields

When proving a theorem using Galois theory it is useful to consider groups and fields separately and then finally use Galois theory to connect the results about the two of them together. Therefore, groups will be considered first before connecting them to fields.

Finite cyclic groups:
Oystein Ore proved that a group is locally-cyclic if and only if it is subalgebra-distributive. For a finite cyclic group $G$ we can go one step further and prove that the lattice of subgroups $Sub(G)$ is the most fundamental type of distributive lattice: a multiset inclusion lattice / divisibility lattice.

Theorem. let $G$ be a cylic group of order $n$. Then $Sub(G)$ is the divisibilty lattice of the factors of $n$. [1]

Cyclic extensions:
In order to relate cyclic groups and their subalgebra lattices to finite fields, we need to show that finite extensions of finite fields are cyclic. This can be shown using the Frobenius automorphism [2] which generates the Galois group.

Theorem. let $F$ be a finite field of order $p^n$, then $Gal(F/GF(p))$ is a cyclic group of order $n$. [2]

Galois theory:
With the group-theoretic prelimanaries out of the way, we can now prove that the subalgebra lattice of a finite field $Sub(F)$ is a multiset inclusion lattice.

Theorem. let $F$ be a finite field, then $Sub(F)$ is a multiset inclusion lattice.

Let $F$ be a finite field of order $p^n$, then the Galois group of $F$ over its prime field is a cyclic group of order $n$. By the fundamental theorem of Galois theory, $Sub(F)$ is therefore order-dual to the subalgebra lattice of a cyclic group of order $n$. This subgroup lattice is a multiset inclusion lattice, but since finite total orders are order-dual so is its order dual. Therefore, the lattice of subfields of the finite field $Sub(F)$ is a multiset inclusion lattice as well.

References:

[1] Subgroup of Finite Cyclic Group is Determined by Order
https://proofwiki.org/wiki/Subgroup_of_Finite_Cyclic_Group_is_Determined_by_Order

[2] The Frobenius Automorphism for a Finite Field
http://mathonline.wikidot.com/the-frobenius-automorphism-for-a-finite-field