The norm $N(a)$ of an $a$ element of a field extension $K/L$ is the determinant of the matrix representing multiplication by $a$. It has the following properties: $$ N(a b) = N(a)N(b) \\ N(ka)=k^n N(a) $$ where $ a,b\in L$ and $k\in K$ and $n$ is the degree of the extension.

But there is no mention of addition.


1 Answer 1


No, the field norm is not a norm in the sense of normed vector spaces.

One reason is that the field norm takes values in $L$ and vector space norms take values in $\mathbb R$.

Even when $L \subset \mathbb R$, the field norm is not a vector space norm because it can be negative.

Wikipedia offers this example: In $\mathbb Q(\sqrt{2})$, the field norm of $ 1+\sqrt{2}$ is $-1$.

  • 1
    $\begingroup$ Indeed, two parts of the definition of a vector space norm wouldn't even make sense unless the field is ordered: $\|x\| \geq 0$ and $\|x + y\| \leq \|x\| + \|y\|$. $\endgroup$
    – user169852
    Oct 8, 2015 at 2:35
  • $\begingroup$ if we take a ordered field L only when norm of field is works as norm on vector space . Am I correct sir ? $\endgroup$
    – Gilll
    Dec 30, 2017 at 3:11

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