If $K$ is a field extension with degree $n=[K:\mathbb{Q}]$ over $\mathbb{Q}$, then there are precisely $n$ field embeddings $\sigma_1,...,\sigma_n$ of $K$ into $\mathbb{C}$, and we define the field norm $N^K_\mathbb{Q}(\alpha)$ for $\alpha\in K$ as the product: $$ N^K_\mathbb{Q}(\alpha) = \prod_{i=1}^n\sigma_i(\alpha) $$ It can be shown that $N^K_\mathbb{Q}(\alpha)\in \mathbb{Q}$ and, in particular, if $\alpha\in \mathcal{O}_K$ then $N^K_\mathbb{Q}(\alpha)\in \mathbb{Z}$. Similarly, if $L$ is a degree $n = [L:K]$ extension over $K$, then there are $n$ field embeddings $\sigma_1,\dots,\sigma_n$ of $L$ into $\mathbb{C}$ that fix $K$ pointwise. We define the relative norm of $N^L_K(\alpha)$ for $\alpha\in L$ in the same way: $$ N^L_K(\alpha) = \prod_{i=1}^n \sigma_i(\alpha) $$ It can be shown in a similar way that $N^L_K(\alpha) \in K$ with $N^L_K(\alpha)\in \mathcal{O}_K$ if $\alpha\in \mathcal{O}_L$.
My question is this: If $L$ is Galois over $K$ (resp. $K$ is Galois over $\mathbb{Q}$) I often see an alternative definition of the norm defined as: $$ N^L_K(\alpha) = \prod_{\sigma\in \text{Gal}(L/K)} \sigma(\alpha) $$ Now, I can believe that the Galois automorphisms of $L$ over $K$ correspond to its field embeddings that fix $K$ pointwise. But how would one show that the two definitions of norm output the same number (i.e. they are actually equivalent)?