Let L = $\mathbb{Q}(\zeta)$ where $\zeta$ is the $n^{th}$ root of unity and K = $\mathbb{Q}$. My definition of the norm is that for any $\alpha \in$ L, $N^L_K(\alpha) = \prod_i \sigma_i(\alpha)$. Now I know that the embeddings of these fields are $\sigma_i:\zeta \rightarrow \zeta^i \; \forall i$ coprime to n. My question is to do with elements that aren't easily sorted into the form $\alpha = a\zeta +b\zeta^2 + ... $.
For example, I am trying to establish whether $2\sqrt5 - 1$ is prime in $\mathbb{Q}(\zeta_5)$. Now I know it is prime in $\mathbb{Q}(\sqrt5)$ as its norm is 19 and therefore prime and it is a member of $\mathbb{Q}(\zeta_5)$ as $\mathbb{Q}(\sqrt5)$ is a subfield of $\mathbb{Q}(\zeta_5)$. I doubt the norm is simply $(2\sqrt5 - 1)^4$ as it is not in $\mathbb{Z}$ and I am not able to figure out the embeddings of $\mathbb{Q}(\zeta_5)$ into $\mathbb{Q}(\sqrt5)$ to use the transitivity of norms to answer my question.
Any help would be much appreciated, especially if you could give some guidance on a general case as well as the specific one stated.