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Suppose $L / K$ is a cyclic extension of number fields. Is there a straightforward way to determine if a given $\alpha\in K^{*}$ is in the image of the norm map $N_{L/K}:L^{*}\rightarrow K^{*}$? Or to at least find an element of $K^{*}$ not in the image?

For example, if $K=\mathbb{Q}(\sqrt{-3})=\mathbb{Q}(\omega)$ where $\omega$ is a primitive third root of 1, and $L=\mathbb{Q}(\omega, \sqrt[3]{2})$ then how might I go about finding an element of $K^{*}$ which is not a norm? I know that $N_{L/K}(a+b\sqrt[3]{2} + c\sqrt[3]{2}^2)=a^2 + 2b^2 + 4c^2 -6abc$ for $a,b,c\in K=Q(\omega)$, but finding an element of $\mathbb{Q}(\omega)$ that can't be written in this form seems impossible to do directly.

I suspect that one answer may involve the Hasse norm theorem and local class field theory, but I know very little about this subject.

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This articles describes an algorithm: https://math.uni-paderborn.de/fileadmin/mathematik/AG-Computeralgebra/Publications-klueners/norms_jsc.pdf (Jürgen Klüners, V. Acciaro: Computing Local Artin Maps, and Solvability of Norm Equations, J.Symb.Comput., 30, 2000, 239-252.)

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