Trace and Norm Generalized

Let \(K, L\) be number fields satisfying \(K \subset L\). Let \(\sigma_1,...,\sigma_n\) be the \(n = [L:K]\) embeddings of \(L\) in \(\mathbb{C}\) that fix \(K\). Let \(\alpha \in L\). Then define the relative trace and relative norm as follows

\[\array { T^L_K(\alpha)=\sigma_1(\alpha)+...+\sigma_n(\alpha) \\ N^L_K(\alpha)=\sigma_1(\alpha)...\sigma_n(\alpha) }\]

Thus \(T^K = T^K_\mathbb{Q}\) and \(N^K = N^K_\mathbb{Q}\). As before we can show:

Theorem: 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

\[\array { T^L_K(\alpha) = \frac{n}{d}t(\alpha) \\ N^L_K(\alpha) = n(\alpha)^\frac{n}{d} }\]

Corollary: \(T^L_K(\alpha), N^L_K(\alpha) \in K\). If \(\alpha\) lies in the number ring of \(L\), then they lie in the number ring of \(K\).

Theorem: Let \(K,L,M\) be number fields with \(K\subset L \subset M\). Then for all \(\alpha \in M\) we have transitivity in the following sense

\[\array { T^L_K(T^M_L(\alpha)) &=& T^M_K(\alpha) \\ N^L_K(N^M_L(\alpha)) &=& N^M_K(\alpha) }\]

Proof: Let \(\sigma_1,...,\sigma_n\) be the embeddings of \(L\) in \(\mathbb{C}\) that fix \(K\), and let \(\tau_1,...,\tau_n\) be the embeddings of \(M\) in \(\mathbb{C}\) that fix \(L\). We first need to extend the embeddings to automorphisms of some field so that we may compose them. Hence fix a normal extension \(N\) of \(\mathbb{Q}\) such that \(M \subset N\). Then all the \(\sigma_i, \tau_i\) may be extended to automorphisms of \(N\). Fix one extension of each and keep the labels \(\sigma_i, tau_i\). Now the mappings can be composed:

\[\array { T^L_K(T^M_L(\alpha)) &=& \sum_{i=1}^{n}\sigma_i{(\sum_{j=1}^m\tau_j(\alpha))} &=& \sum_{i,j}\sigma_i\tau_j(\alpha) \\ N^L_K(N^M_L(\alpha)) &=& \prod_{i=1}^{n}\sigma_i{(\prod_{j=1}^m\tau_j(\alpha))} &=& \prod_{i,j}\sigma_i\tau_j(\alpha) }\]

We now need to show that the \(mn\) mappings \(\sigma_i \tau_j\) restricted to \(M\) are the embeddings of \(M\) in \(\mathbb{C}\) which fix \(K\). Since all \(\sigma_i\tau_j\) fix \(K\) and there are \(mn = [M:L][L:K] = [M:K]\) of them, it remains to show they are all distinct when restricted to \(M\).

Suppose two of the mappings agreed on \(M\). Then they also agree on \(L\). The \(\tau\) maps fix \(L\), so this means we have \(\sigma_i \ne \sigma_j\) agreeing on all of \(L\), which is a contradiction.∎

We may interpret the trace an norm as follows. Let \(K \subset L\) be fields and let \(\alpha \in L\). Then considering \(L\) as a vector space over \(K\), multiplication by \(\alpha\) is a linear mapping. Let \(A\) be a matrix representing this map with respect to some basis \(\alpha_1,\alpha_2,...\) for \(L\) over \(K\). Then \(T^L_K(\alpha)\) and \(N^L_K(\alpha)\) are the trace and determinant of \(A\).

Ben Lynn 💡