## 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.

Ben Lynn blynn@cs.stanford.edu 💡