On p.17 of Number Fields by Marcus, the following Theorem is given:
Let $\alpha \in L$ and let $d$ be the degree of $\alpha$ over $K$. Let $t(\alpha)$ and $n(\alpha)$ be the sum and product of the $d$ conjugates of $\alpha$ over $K$. Then $$T_K^L(\alpha) = \frac{n}{d}t(\alpha)$$ $$N_K^L(\alpha) = n(\alpha)^{\frac{n}{d}}.$$
A corollary to this is:
$T_K^L(\alpha)$ and $N_K^L(\alpha)$ lie in $K$. If $\alpha \in \mathcal{O}_L$ then $T_K^L(\alpha)$ and $N_K^L(\alpha)$ lie in $\mathcal{O}_K$.
The corollary is stated without proof and this is likely due to the proof being very similar to that of one from earlier. The first part of the corollary is clear since the relative trace and norm are (nearly) coefficients of the minimal polynomial of $\alpha$ over $K$ and hence each lies in $K$ but what isn't clear to me is the second part of the corollary.
How would one prove the second part of the corollary? If $K = \mathbb{Q}$ then the result is clear since the minimal polynomial would lie in $\mathbb{Z}[x]$ but I don't see how that argument extends to cases where $K \neq \mathbb{Q}.$
Any help is appreciated!
Best Answer
There are many different ways of defining Norm and Trace. Once you get very familiar with these matters, you may find it instructive to show that they’re all equivalent.
My favorite definition makes it obvious that the Norm and Trace land you in the base ring(field), but it is not obvious that it’s the same as what you’ve seen. Here it is:
Consider your extension field $L$ over $K$, and choose any basis that you like, say $\{b_1,\cdots,b_n\}$. Then for a given element $a\in L$, define $\tau_a:L\to L$ as, for $z\in L$, $\tau_a(z)=az$. You check that $\tau_a$ is a $K$-linear transformation $L\to L$, and therefore it has a Determinant and a Trace. These are $\mathbf N^L_K(a)$ and $\mathbf{Tr}^L_K(a)$, respectively. If you like, you can use the basis above to represent $\tau_a$ as an $n$-by-$n$ matrix over $K$, and then the Norm and Trace are, again, the Determinant and the Trace of this matrix.
This is called the regular representation of $L$: you’ve represented the elements of $L$ as linear endomorphisms of the $K$-space $L$.