AlgebraicNumberNorm

AlgebraicNumberNorm[a]

gives the norm of the algebraic number a.

Details and Options

  • The norm of a is defined to be the product of the roots of its minimal polynomial.
  • AlgebraicNumberNorm[a,Extension->θ] finds the norm of a over the field .

Examples

open allclose all

Basic Examples  (1)

Norms of algebraic numbers:

Scope  (4)

Integers and rational numbers:

Radical expressions:

Root and AlgebraicNumber objects:

AlgebraicNumberNorm automatically threads over lists:

Options  (1)

Extension  (1)

Norm of over :

Applications  (1)

is irreducible in :

Since AlgebraicNumberNorm is multiplicative, having a prime norm implies the original number is prime:

Properties & Relations  (3)

AlgebraicNumberNorm is multiplicative:

Units in a number field have norm :

Compute the smallest field that includes , i.e. :

Compute the norm in that field:

Neat Examples  (1)

Plot of norms of elements in :

Introduced in 2007
 (6.0)