Theorem 7: Let p be a prime number and m,n be positive integers. ThenF is a subfield of F. if and only if mln.Proof: Suppose, first, that F is a subfield of F.. Then there exists aninjective 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 isp²not clearRequest explainthe proofNow [FF] n and [E: F₂]=[FF] = m.=So [FF] [FEE: F₂]i.e., n=[F.: Flm,i.e., mln.

Answered: Image /qna-images/answer/3ed9fb57-f424-4a1f-95f1-3fe3a267d58e.jpg

Answered: Theorem 7: Let p be a prime number and…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.3: The Characteristic Of A Ring

Problem 7E

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