# Galois Extension

Not what you're looking for?

** 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.

##### Purchase this Solution

##### Solution Summary

This solution two algebra questions regarding Galois extension.

##### Purchase this Solution

##### Free BrainMass Quizzes

##### Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

##### Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

##### Probability Quiz

Some questions on probability

##### Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

##### Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.