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. 



Determining the image of the norm map of a cyclic extension
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.)


