> That part is fine. But why, then, does multiplying zzˉ give a “magnitude” that works in a reasonable way?
Because the product of all Galois conjugates is a norm and the determinant of the linear operator defined by general field multiplication of a primitive element when viewing the field extension as a vector space of the extension field over the base field.
Although the geometric interpretation of norms in Galois theory really only works for the complex numbers because only the complex numbers are a field. Quaternions are not a field.
Because the product of all Galois conjugates is a norm and the determinant of the linear operator defined by general field multiplication of a primitive element when viewing the field extension as a vector space of the extension field over the base field.
Although the geometric interpretation of norms in Galois theory really only works for the complex numbers because only the complex numbers are a field. Quaternions are not a field.