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Upgrading from:

NumberTheory`AlgebraicNumberFields`

Algebraic is now available as the newly added built-in Mathematica kernel function AlgebraicNumber.
ToCommonField and ToNumberFieldElement are now available as the newly added built-in Mathematica kernel function ToNumberField.
MinimalPolynomial, AlgebraicIntegerQ, AlgebraicNumberDenominator, AlgebraicNumberTrace, AlgebraicNumberNorm, AlgebraicUnitQ, and RootOfUnityQ have been added to the built-in Mathematica kernel.
IntegralBasis is now available as the newly added built-in Mathematica kernel function NumberFieldIntegralBasis.
FundamentalUnits is now available as the newly added built-in Mathematica kernel function NumberFieldFundamentalUnits.
NormRepresentatives is now available as the newly added built-in Mathematica kernel function NumberFieldNormRepresentatives.
NumberFieldSignature, NumberFieldDiscriminant, and NumberFieldRegulator have been added to the built-in Mathematica kernel.
RootsOfUnity is now available as the newly added built-in Mathematica kernel function NumberFieldRootsOfUnity.

AlgebraicNumber

Represent an algebraic number and get a numerical approximation:

Version 5.2 << NumberTheory`AlgebraicNumberFields`; a = Algebraic[#1^3 - #1 + 1 &, {1, 2, 3}, 1]; N[a, 20]
Wolfram Language code: a = AlgebraicNumber[Root[#1^3 - #1 + 1&, 1], {1, 2, 3}]
Wolfram Language code: N[a, 20]

ToNumberField

This finds a common finite extension of rationals containing the given algebraic numbers:

Version 5.2 << NumberTheory`AlgebraicNumberFields`; ToCommonField[{Sqrt[2], Sqrt[3]}]
Wolfram Language code: ToNumberField[{Sqrt[2], Sqrt[3]}]

This finds the smallest common finite extension of rationals containing the given algebraic numbers:

Version 5.2 ToCommonField[{Algebraic[1 - 10 #1^2 + #1^4 &, {0, -9/2, 0, 1/2}, 4], Sqrt[5]}, SmallestField -> True]
Wolfram Language code: ToNumberField[{AlgebraicNumber[Root[1 - 10 #1 ^ 2 + #1 ^ 4&, 4], {0, -9 / 2, 0, 1 / 2}], Sqrt[5]}, All]

This represents as an element of the field generated by Root[1-10#12+#14&,4]:

Version 5.2 ToNumberFieldElement[Sqrt[6], 1 - 10 #1^2 + #1^4 &, 4]
Wolfram Language code: ToNumberField[Sqrt[6], Root[1 - 10 #1^2 + #1^4&, 4]]

MinimalPolynomial

This gives the minimal polynomial of expressed as a pure function:

Version 5.2 << NumberTheory`AlgebraicNumberFields`; MinimalPolynomial[Sqrt[2] + Sqrt[3]]
Wolfram Language code: MinimalPolynomial[Sqrt[2] + Sqrt[3]]

NumberFieldIntegralBasis

This gives an integral basis of the field generated by the first root of 533+429 #1+18 #12+#13&:

Version 5.2 << NumberTheory`AlgebraicNumberFields`; IntegralBasis[533 + 429 #1 + 18 #1^2 + #1^3 &, 1]
Wolfram Language code: NumberFieldIntegralBasis[533 + 429 #1 + 18 #1^2 + #1^3&, 1]

NumberFieldFundamentalUnits

This gives a fundamental unit of the field [Sqrt[2]] generated by the algebraic number Sqrt[2]:

Version 5.2 << NumberTheory`AlgebraicNumberFields`; FundamentalUnits[Sqrt[2]]
Wolfram Language code: NumberFieldFundamentalUnits[Sqrt[2]]

NumberFieldNormRepresentatives

This gives a set of representatives of classes of elements of norm 2 in the field generated by Sqrt[2]+ Sqrt[3]:

Version 5.2 << NumberTheory`AlgebraicNumberFields`; NormRepresentatives[2, Sqrt[2] + Sqrt[3]]
Wolfram Language code: NumberFieldNormRepresentatives[Sqrt[2] + Sqrt[3], 2]

NumberFieldSignature

Find the signature of the number field :

Version 5.2 << NumberTheory`AlgebraicNumberFields`; NumberFieldSignature[Sqrt[2] + Sqrt[3]]
Wolfram Language code: NumberFieldSignature[Sqrt[2] + Sqrt[3]]

NumberFieldRootsOfUnity

Here are all roots of unity in the field :

Version 5.2 << NumberTheory`AlgebraicNumberFields`; RootsOfUnity[1 + I Sqrt[3]]
Wolfram Language code: NumberFieldRootsOfUnity[1 + I Sqrt[3]]

NumberFieldCharacteristicPolynomial and NumberFieldTrace were available in previous versions of Mathematica and are now available on the web at library.wolfram.com/infocenter/MathSource/6827.