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].

See also

Determinant, Field Trace, Finite Extension, Norm, Number Field, Ring of Integers, Unit

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References

Milne, J. S. "Algebraic Number Theory." 19 Jul 2020. https://www.jmilne.org/math/CourseNotes/ANT.pdf.

Cite this as:

Weisstein, Eric W. "Field Norm." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FieldNorm.html

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