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:

$N(x)=(-1{)}^d{a}_0^{d/d_1}$.

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. For full proof, refer: algebraic norm in a number field is multiplicative
• The norm is nonzero on all nonzero elements.