Field norm

The field norm $N_{L/K}(x)$ is the determinant of multiplication by $x$, equivalently the product of its conjugates when the extension is separable.

For a finite number field $L/\mathbb{Q}$ with a rational basis ${\omega }_1,\dots ,{\omega }_n$, the norm form is the homogeneous polynomial

If $x=y^2$ in a number field, then $N(x)=N(y{)}^2$; a norm that is not a square in the base field therefore obstructs square roots.

An algebraic integer whose absolute norm is $1$ or $-1$ is a unit, because its conjugate product supplies an integral inverse.