Number Fields

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 $ℂ$ where $n=[K:\mathbb{Q}]$. For $a\in K$, define

$$\begin{array}{ccc}T(a)& =& {\sigma }_1(a)+...+{\sigma }_n(a)\\ N(a)& =& {\sigma }_1(a)...{\sigma }_n(a)\end{array}$$

(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:

$$\begin{array}{ccc}T(a)& =& \frac{n}{d}t(a)\\ N(a)& =& n(a{)}^{\frac{n}{d}}\end{array}$$

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 $ℂ$ extends to $n/d$ embeddings of $K$ in $ℂ$.

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.