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163 views

Trace and cyclotomic field

Let $K=\mathbb{Q}(\zeta_p)$ be the cyclotomic field of $p$th roots of unity for the prime $p$ and let $G=\operatorname{Gal}(K/\mathbb{Q})$. Let $\zeta$ denote any $p$th root of unity. Please show that ...
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In a quadratic extension, does the orthogonal complement of trace have a nice name?

The trace of an element in a quadratic extension is the sum of it with its conjugate. Is there a name for the difference between it and its conjugate?