(a) Prove: If H is a subfield of GF(pn), then H has order pm for some divisor m of

Chapter 50, Problem 50.10

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(a) Prove: If H is a subfield of GF(pn), then H has order pm for some divisor m of n.[Suggestion: What does the proof of Theorem 50.2 say about [H : Zp]? A similar ideaapplies to [GF(pn) : H].](b) Prove: If m and n are positive integers and min, then (pm - I) I (p" - 1).(c) Prove: If min, then GF(pn) has a subfield of order pm.

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