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Irreducible Polynomials, Powers and Primes

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1.-Let p a prime and let , (this is a extension field) , where is an irreducible polynomial over . Show that if are elements of that satisfy .
Note this show that the pth powers of the elements of are distinct, and therefore every element in is the pth power of a unique element in . Therefore every element in has a unique pth root; that is, for any in there is a unique in such that . This property is not true for the field of real numbers since .

Please, can you explain with details this problem by step by step?
Remember that: Let a and b be indeterminants over the field , where p is prime then

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Irreducible Polynomials, Powers and Primes are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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