Discussion
Let K/F be a Galois extension of fields. If Gal(K/F) is isomoprhic to \(S_{3}\), the number of fields L such that \(F \subseteq L \subseteq K\) is :
Answer: A
There are 6 intermediate fields.
By the fundamental theorem of Galois theory, the fields L with \(F \subseteq L \subseteq K\) correspond one-to-one with the subgroups of Gal(K/F). \(S_{3}\) has exactly six subgroups:
- the trivial subgroup (giving L = K)
- three subgroups of order 2, generated by the transpositions
- the alternating group \(A_{3}\) of order 3
- \(S_{3}\) itself (giving L = F)
So there are 6 fields, counting F and K themselves. 4 forgets the two trivial ends, and 8 or 10 overcount the subgroups.
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