What is meant by the relative trace?

Question

My lecturer of Algebraic Number Theory was talking about the relative trace, but I have no idea what this means. For example he was talking about $\mathbb{Q}(\sqrt(6),i)$ with the subfield $\mathbb{Q}(i)$ and says that for an integral element $\alpha=p+qi+-r\sqrt(6)-s\sqrt(-6)$ the relative trace of $\alpha$ in $\mathbb{Q}(i)$ is $2p+2qi \in \mathbb{Z}[i]$. How do I calculate this trace and how do I know it is in $\mathbb{Z}[i]$?

Answer 1

If $L/K$ is an extension of number fields, and $\alpha \in L$, then there is a $K$-linear map given by $$m_\alpha: x \mapsto \alpha x$$

The relative trace, $\mathrm{Tr}_{L/K}(\alpha)$ is defined to be the regular linear algebraic trace of the linear map $m_\alpha$. Since traces are scalar valued, the trace will lie in $K$.

We can show using linear algebra that $$\mathrm{Tr}_{L/K}(\alpha)=\sum_{\sigma:L\hookrightarrow \mathbb C}\sigma(\alpha)$$ where the sum is over $K$-embeddings into $\mathbb C$. (Think of the values $\sigma(\alpha)$ as other roots of the minimal polynomial of $\alpha$.)

In particular, if $\alpha \in \mathcal O_L$ is an algebraic integer, then $\mathrm{Tr}_{L/K}(\alpha)\in\mathcal O_L\cap K = \mathcal O_K$.