I am studying Galois Theory and we were introduced to the concept of a field norm, which is defined as follows (taken from Wikipedia):
" Let $K$ be a field and $L$ a finite extension of $K$. The field $L$ is then a finite dimensional vector space over $K$.
Multiplication by $\alpha \in L$ given by the map: $$m_\alpha:L\to L $$ $$ x \mapsto \alpha \cdot x $$
is a $K$-linear transformation of this vector space into itself.
The norm $N_{L / K}(\alpha)$ is defined as the determinant of this linear transformation. "
I understand that the field norm is not a "regular" norm in a vector space, and also in Wikipedia it says that "the field norm is very different from the usual distance norm" and an example is given as well. I also saw some more examples distinguishing the field norm from the "usual distance norm".
So, if it is different why call it a norm? Is there any specific case where this field norm actually is related to a norm in a vector space?
I looked around quite a bit for an answer but could not find one, I hope this isn't a duplicate.
Thanks in advance!
Yes! Consider the finite extension $\mathbb{C}/\mathbb{R}$. Then the norm map $N_{\mathbb{C}/\mathbb{R}} : \mathbb{C} \to \mathbb{R}$ is the square of the standard norm, that is $$N_{\mathbb{C}/\mathbb{R}}(a + bi) = a^2 + b^2.$$
To see this, consider the ordered basis $(1,i)$ for $\mathbb{C}$. The matrix representation of $m_{a + bi}$ with respect to this ordered basis is $$\begin{bmatrix} a & -b \\ b & a\end{bmatrix}$$ which has determinant $a^2 + b^2$.