The field norm of an element 
of a finite extension 
is the determinant
 |
where 
is the 
-linear transformation 
on 
, regarded as a vector space
over 
. The field norm is multiplicative, so
(Milne
2020, pp. 31-32). For example, in the quadratic
field 
,
 |
for rational numbers 
and 
,
so 
has field norm 
.
For a number field 
, the field norm 
of an algebraic
integer 
is an integer. An algebraic
integer is a unit in the ring
of integers of 
iff its field norm is 
(Milne 2020, p. 86).
For an algebraic number 
and a positive integer 
, NumberFieldNormRepresentatives[a,
m] returns representatives of the classes of algebraic
integers in 
with field norm 
or 
. Two such algebraic
integers are in the same class when their quotient
is a unit in the ring of
integers of 
.
Multiplying the representatives by these units gives all
algebraic integers with the specified field
norms.
For an algebraic number 
generating 
, the field norm 
is computed in the Wolfram
Language as AlgebraicNumberNorm[a].
