This document discusses a finite field extension L over a field F, where L = F[a] for some element a in L. It shows that the number of intermediate fields between F and L, denoted F(L/F), is finite by the following logic: Since L is a finite extension of F generated by a single element a, the degree of the extension F(a)/F is at least 1. Therefore, the number of intermediate fields F(L/F), which is equal to F(F(a)/F), is bounded above by 1 and is thus finite rather than infinite.

1. Let F <= L be a finite extension of fields, and let L = F[a]
for some a in L, Show that |F(L/F)| < infinity.
where F(L/F) is the set of intermediate fields between F and L
Solution
given.
F<=L
so,
F/L <= 1
L/F >=1
given L= F(a)
hence
F(a)/ F >=1

2. F (L/F)
= F ( F(a)/F))
but
F(a)/ F >=1
hence
it is limited to 1 and greter than that
so that we confidently say that it is less than infinite