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Galois Extension

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** Please see the attachment for a full problem description **

Suppose that K/F is a Galois extension, the Norm of alpha is an element of K is N_K/F := (please see the attached file) and trace is Tr_K/F(alpha) = (please see the attached file). Suppose also (please see the attached file) that Gal(K/F) is cyclic and K, F are finite fields. Let F = F_p whose order is prime, K = F_q, q = p^k.
a_ Show that Tr_K/F(alpha) = alpha + alpha^p + alpha^p^2 + ... + alpha^p^k-1 and using the other results you may know about trace show that Tr_K/F : K - F is an F -linear map.

b) For alpha is an element of K, show that Tr_K/F(alpha) = 0 if, and only if alpha = (please see the attached file), for some (please see the attached file) is an element of K.

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Solution Summary

This solution two algebra questions regarding Galois extension.

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