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1 Let p be a prime, and consider the ficld Zp a) Show that, if p 2, not cvery clement of Z is the square of some other clement.[Hint: 12 = (p -1)2 = 1. Now count the rcst.] b) Using part (a) where necessary, show that there exist finite fields of order p2 for every prime number p (even p = 2). 2 Let F be a field of characteristic p. Recall from Homework 11 that the Frobenius map ,: F > F $p() = oP, is a ring homomorphism. Use this fact to show that the polynomial :rP - 1 F[r] is reducible. What are its roots?

          1 Let p be a prime, and consider the ficld Zp
a) Show that, if p  2, not cvery clement of Z is the square of some other clement.[Hint: 12 = (p -1)2 = 1. Now count the rcst.]
b) Using part (a) where necessary, show that there exist finite fields of order p2 for every prime number p (even p = 2).
2 Let F be a field of characteristic p. Recall from Homework 11 that the Frobenius map ,: F > F $p() = oP, is a ring homomorphism. Use this fact to show that the polynomial :rP - 1  F[r] is reducible. What are its roots?

1 let p be a prime and consider the ficld zp a show that if p 2 not cvery clement of z is the square of some other clementhint 12 p 12 1 now count the rcst b using part a where necessary sho 57885

Added by Rocio R.

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