The Trace And The Norm

Let $K$ be a number field. Define the trace $T^K$ and the norm $N^K$ as follows. Let $\sigma_1,...,\sigma_n$ be the embeddings of $K$ in $\mathbb{C}$ where $n = [K:\mathbb{Q}]$. For $a \in K$, define

\[ \array { T(a) & = & \sigma_1(a) +...+ \sigma_n(a) \\ N(a) & = & \sigma_1(a) ... \sigma_n(a) } \]

(we omit the field when it is clear which one we are referring to).

For the next part, let $a$ have degree $d$ over $\mathbb{Q}$. Let $t, n$ denote the sum and product of the $d$ conjugates of $a$ over $\mathbb{Q}$.

Theorem:

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

where $n = [K:\mathbb{Q}]$. (Note $n / d = [K:\mathbb{Q}[a]]$).

Proof: $t, n$ are simply $T^{\mathbb{Q}[a]}, N^{\mathbb{Q}[a]}$, and each embedding of $\mathbb{Q}[a]$ in $\mathbb{C}$ extends to $n /d$ embeddings of $K$ in $\mathbb{C}$.

Corollary: $T(a), N(a) \in \mathbb{Q}$.

Proof: This follows from the fact that $-t(a), \pm n(a)$ are coefficients of the minimal polynomial of $a$.

Corollary: $T(a), N(a) \in \mathbb{Z}$ if $a$ is an algebraic integer.

For example, in the quadratic field $K = \mathbb{Q}[\sqrt{m}]$ we have $T(a + b\sqrt{m}) = 2a, N(a + b\sqrt{m}) = a^2 - m b^2$ (where $a,b\in\mathbb{Q}$). For this case, the converse of the corollary is also true.