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Math Help - Find trace

  1. #1
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    Find trace

    Let p be prime, and let K = \mathbb {Q} [w] , w is a primitive pth root of unity. Compute Tr_{K}(w) and  Tr_{K} (1-w)

    Also, show that  (1-w)(1-w^2)... (1-w^{p-1}) = p

    proof.

    So I have  w^p = 1 with  w^{k} \neq 1 \ \ \ \forall k < p
    Last edited by tttcomrader; March 8th 2008 at 02:09 PM.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Let p be prime, and let K = \mathbb {Q} [w] , w is a primitive pth root of unity. Compute Tr_{K}(w) and  Tr_{K} (1-w)
    If E/F is a finite Galois extension then we define \text{Tr}(\alpha) = \sum_{\theta \in \text{Gal}(E/F)}\theta (\alpha) for \alpha \in E. Now if \omega is a primitive p-th root of unity then \mathbb{Q}(\omega) is finite and Galois over \mathbb{Q}, furthermore, since \{ 1,\omega, ... , \omega^{p-1}\} for a basis for this extension and \mathbb{Q} is a field with \text{Char}(\mathbb{Q}) = 0 it means G=\text{Gal}(K/\mathbb{Q}) is a group of order p, so it must be a cyclic group. Thus, there is an automorphism \theta of K leaving \mathbb{Q} fixed so that \left< \theta \right> = G. Suppose that \theta (\omega) = \omega^k where 0< k \leq p-1. Then it means \{ \text{Id},\theta, \theta^2, ... ,\theta^{p-1} \} = G. And so \text{Tr}(\omega) = \text{Id}(\omega) + \theta(\omega) + \theta^2 (\omega) + ... + \theta^{p-1} (\omega). Thus, \mbox{Tr}(\omega) = \omega + \omega^k + \omega^{2k} + ... + \omega^{(p-1)k}= \omega - 1.
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