Theorem 7: Let p be a prime number and m,n be positive integers. Then F is a subfield of F. if and only if mln. Proof: Suppose, first, that F is a subfield of F.. Then there exists an injective homomorphism 8: F→F So F Im = E, say. Now, E is a finite subfield of F.. Since the characteristic of F., is p. F, SEC. The proof is p² not clear Request explain the proof Now [FF] n and [E: F₂]=[FF] = m. = So [FF] [FEE: F₂] i.e., n=[F.: Flm, i.e., mln.

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