For complex numbers, $|z| =\sqrt{N(z)}$, where $N(z) =\bar{z}z =\operatorname{det}(m_z)$ is the algebraic norm of $z$ over the reals ($m_z$ is the operator of multiplicaiton by $z$).
The algebraic norm is defined for arbitrary finite field extensions. Does there exist a corresponding general definition of modulus/absolute value?