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

ToNumberField

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

Version 5.2

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

Version 5.2

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

Version 5.2

MinimalPolynomial

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

Version 5.2

NumberFieldIntegralBasis

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

Version 5.2

NumberFieldFundamentalUnits

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

Version 5.2

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

NumberFieldSignature

Find the signature of the number field :

Version 5.2

NumberFieldRootsOfUnity

Here are all roots of unity in the field :

Version 5.2

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