Algebraic norm in a number field

Definition
The algebraic norm in a number field is a map from the number field to the field of rational numbers, defined as follows. If the number field has degree $$d$$, the minimal polynomial of a given element $$x$$ has degree $$d_1$$, and the constant term of its minimal monic polynomial is $$a_0$$, we define:

$$N(x) = (-1)^da_0^{d/d_1}$$.

Notice that this is not an integer-valued function on a number field; however, its restriction to the ring of integers is an integer-valued function, and hence a norm on a commutative unital ring. However, that norm need not necessarily be a nonnegative norm.

Facts

 * The algebraic norm in any number field is multiplicative: the norm of a product of elements equals the product of their norms.
 * The norm is nonzero on all nonzero elements.