Triangular inequality for the Norm in the field extension

Question

Let K(a) be a field extension of field K with algebraic number a. It is well known that Norm(bc)=Norm(b)Norm(c) for b and c in K(a). It can be proved by the property of polynomial resultant. In general, the term "Norm" should satisfy the triangle inequality. Is it true in the field extension case? thanks

Answer 1

The meanings of the word "Norm" in algebra and analysis are quite different. The norm in fields does not generally satisfy the triangle inequality, even in an ordered field. For example, the norms of $1\pm\sqrt{2}$ (over the rationals) are $-1$, while the norm of $2 = (1+\sqrt{2}) + (1-\sqrt{2})$ is $4$.

Answer 2

No in general. For example, $K=\mathbf{Q}(\sqrt{2})$, $a,b\in \mathbf{Z}$ a solution to Pell equation: $a^2-2b^2=1$ with $b$ large enough. Take $x=a+b\sqrt{2}$ and $y=-a+b\sqrt{2}$. Then $x+y=2b\sqrt{2}$, thus $|N(x+y)|=8b^2$, while $N(x)=N(y)=1$. So, we cannot control $|N(x+y)|$ with $|N(x)|,|N(y)|, K$.