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NumberFieldNormRepresentatives

NumberFieldNormRepresentatives

NumberFieldNormRepresentatives[a,m]

gives a list of representatives of classes of algebraic integers of norm in the field generated by the algebraic number .

Details

Algebraic integers are considered to be in the same class if their quotient is a unit in the field .
All elements of the number field with norm can be obtained from the representatives by multiplication by units in the field.

Examples

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Basic Examples (1)

Find the representatives of classes of algebraic integers of norm in :

Wolfram Language code: NumberFieldNormRepresentatives[Sqrt[2], 7]

Wolfram Language code: AlgebraicNumberNorm[%]

Scope (4)

Radical expressions:

Wolfram Language code: NumberFieldNormRepresentatives[Sqrt[2] + Sqrt[3], 2]

Root objects:

Wolfram Language code: NumberFieldNormRepresentatives[Root[2 + 3#1 ^ 3&, 2], 3]

AlgebraicNumber objects:

Wolfram Language code: NumberFieldNormRepresentatives[AlgebraicNumber[Root[2 + #1 ^ 3&, 1], {-1, -1, 2}], 12]

NumberFieldNormRepresentatives automatically threads over lists:

Wolfram Language code: NumberFieldNormRepresentatives[{Sqrt[3], E ^ (Pi * I / 4)}, 2]

Properties & Relations (5)

Representatives of norm in :

Wolfram Language code: {a} = NumberFieldNormRepresentatives[Sqrt[5], 5]

Wolfram Language code: AlgebraicNumberNorm[a]

The number has norm :

Wolfram Language code: AlgebraicNumberNorm[Sqrt[5]]

It can be represented in terms of the representative by multiplying by a unit:

Wolfram Language code: RootReduce[Sqrt[5](-2 - Sqrt[5]) == a]

Wolfram Language code: AlgebraicUnitQ[-2 - Sqrt[5]]

Obtain all elements of norm in by multiplying representatives with units:

Wolfram Language code: {a, b} = NumberFieldNormRepresentatives[Sqrt[2], 7]

Wolfram Language code: {u} = NumberFieldFundamentalUnits[Sqrt[2]]

Elements generated by :

Wolfram Language code: Table[RootReduce[a u ^ n], {n, 10}]

Wolfram Language code: AlgebraicNumberNorm /@ %

Elements generated by :

Wolfram Language code: Table[RootReduce[b u ^ n], {n, 10}]

Wolfram Language code: AlgebraicNumberNorm /@ %

FindInstance gives all Gaussian integers of norm :

Wolfram Language code: gi1 = a + I b /. FindInstance[a ^ 2 + b ^ 2 == 5, {a, b}, Integers, 100]

Check the result:

Wolfram Language code: NumberFieldNormRepresentatives[I, 5]

Wolfram Language code: gi2 = Flatten @ Outer[Times, {-1, 1, -I, I}, RootReduce[%]]

Wolfram Language code: Sort[gi1] == Sort[gi2]

Find an instance of a quadratic equation :

Wolfram Language code: First @ RootReduce @NumberFieldNormRepresentatives[Sqrt[2], 8]

Wolfram Language code: {a, b} = % /. a_ + Sqrt[2]b_ :> {a, b}

Wolfram Language code: a ^ 2 - 2b ^ 2

Find the representatives of classes of algebraic integers of norm in :

Wolfram Language code: primitive = ToNumberField[{Sqrt[2], Sqrt[3]}, All][[1, 1]]

Wolfram Language code: NumberFieldNormRepresentatives[primitive, 2]

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