Field extensions of prime degree have no intermediate subfields

Let K/F be a field extension with prime degree p. Show that any subfield of K containing F is either K or F.


Let F \subseteq E \subseteq K. Then [K:E][E:F] = [K:F] = p by Theorem 14 in D&F. Since the degree of a (finite) field extension is an integer, either [K:E] = 1 (so E = K) or [E:F] = 1 (so E = F).

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