Algebraic integers, traces, and norms

For a number field K of degree n and $\alpha\in K$, multiplication $m_\alpha:x\mapsto\alpha x$ is a Q-linear endomorphism of the n-dimensional vector space K. Define the field trace as its matrix trace and the field norm as its determinant. Different rational bases give conjugate matrices, so these values are basis independent. The trace is additive and Q-linear, while the norm is multiplicative. The norm is not generally additive, and replacing a determinant by the product of displayed coordinates has no basis-independent meaning.

Every number field is separable over Q, so there are n distinct embeddings into the complex numbers. After extending scalars, multiplication diagonalizes according to those embeddings, making trace their sum and norm their product. Complex embeddings occur in conjugate pairs when the base field is Q. Their paired norm contribution is nonnegative real, but real embeddings can contribute signs. Therefore a nonzero algebraic integer's rational norm can be negative; taking its absolute value is a separate operation used for ideal norms and lattice volumes.

Let F=Q(α) have degree d and let K contain F with degree r over F. The characteristic polynomial of multiplication by α on K is its degree-d minimal polynomial raised to the power r. Thus its field trace in K is r times its trace in F, and its norm in K is the r-th power of its norm in F. Reading only the constant term of the minimal polynomial is therefore insufficient when the requested norm is relative to a larger ambient field.

Trace and norm are also transitive in field towers, with the appropriate relative versions at each stage. A relative norm lies in the intermediate field before its rational norm is taken. Matrix dimensions and polynomial degrees provide easy checks for accidental mixing: a degree-four ambient multiplication map must have a degree-four characteristic polynomial even when the element satisfies a quadratic equation. Repeated roots in that characteristic polynomial do not contradict separability of the field extension; they reflect an element that does not generate the entire field.

Multiplication by an algebraic integer preserves the integer ring. In an integral basis its matrix has integer entries, so its trace and determinant are integers. The converse statement that arbitrary integer trace and norm imply integrality is safe in a quadratic field, where they give the entire monic polynomial, but not as a general high-degree criterion. More characteristic coefficients are involved in higher degree. One should use the full minimal-polynomial or integral-extension criterion rather than two numerical tests outside their valid range.

For an element of the full integer ring, being a unit is equivalent to having rational norm plus or minus one. One direction follows by multiplicativity and integral norms of an inverse. For the other, the monic characteristic equation with constant term a unit expresses an integral inverse as a polynomial in the element. Without the hypothesis of integrality, norm plus or minus one alone does not establish membership in the unit group of the integer ring. Principal ideal norm is the absolute field norm, so it records the same magnitude while forgetting its sign.