What Are the Subfields of F=F_{p^{18}} and Their Lattice Representation?

[mathsss2]

Identify all the subfields of a field F=F_{p^{18}}, with p^{18} elements where p is a prime. Draw the lattice of all subfields.

I am allowed to just use a theorem, but I don't know one to use.

 

[Office_Shredder]

Start with Lagrange's theorem (which is for groups, but all fields are additive groups) to get that the order of any subgroup (and hence the order of any subfield) divides p¹⁸. This narrows down the possible orders of the fields you're looking for

 

[adriank]

Also, if you have a subfield K, its group of units K* is a subgroup of F* as well.

Don't you have a theorem that says the subfields of F_pⁿ are exactly F_p^m where m divides n?

This stuff belongs in the Abstract Algebra forum.