Algebraic Number Fields
Mathematica provides representation of algebraic numbers as Root objects. A Root object contains the minimal polynomial of the algebraic number and the root number—an integer indicating which of the roots of the minimal polynomial the Root object represents. This allows for unique representation of arbitrary complex algebraic numbers. A disadvantage is that performing arithmetic operations in this representation is quite costly. That is why Mathematica requires the use of an additional function, RootReduce, in order to simplify arithmetic expressions. Restricting computations to be within a fixed finite algebraic extension of the rationals, [], allows a more convenient representation of its elements as polynomials in .
AlgebraicNumber[,{c_{0},c_{1},...,c_{n}}]  represent the algebraic number c_{0}+c_{1} +...+c_{n} ^{n} in [] 
Representation of algebraic numbers as elements of a finite extension of rationals.
For any algebraic number and any list of rational numbers {c_{0}, ..., c_{l}} , AlgebraicNumber[, {c_{0}, ..., c_{l}}] evaluates to AlgebraicNumber[, {d_{0}, ..., d_{m}}], such that =d, d is a factor of the leading coefficient of MinimalPolynomial of , such that is an algebraic integer, m is the degree of MinimalPolynomial of , and
AlgebraicNumber automatically makes the generator of the extension an algebraic integer and the coefficient list equal in length to the degree of the extension.
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ToNumberField[a,]  express the algebraic number a in the number field generated by 
ToNumberField[{a_{1},a_{2},...},]  express the a_{i} in the field generated by 
ToNumberField[{a_{1},a_{2},...}]  express the a_{i} in a common extension field generated by a single algebraic number 
Representing arbitrary algebraic numbers as elements of algebraic number fields.
ToNumberField can be used to find a common finite extension of rationals containing the given algebraic numbers.
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This represents as an element of the field generated by Root[110 #1^{2}+#1^{4}&, 4].
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Arithmetic within a fixed finite extension of rationals is much faster than arithmetic within the field of all complex algebraic numbers.
Suppose you need to find the value of rational function f with {x, y, z} replaced by algebraic numbers {a, b, c}. 
A direct computation of the value of f at {a, b, c} using RootReduce takes a rather long time.
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A faster alternative is to do the computation in a common algebraic number field containing {a, b, c}.
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Arithmetic within the common number field is much faster.
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ToNumberField[{a_{1}, a_{2}, ...}] is equivalent to ToNumberField[{a_{1}, a_{2}, ...}, Automatic], and does not necessarily use the smallest common field extension. ToNumberField[{a_{1}, a_{2}, ...}, All] always uses the smallest common field extension.
Here the first AlgebraicNumber object is equal to so it does not generate the 4 ^{th}degree field ( Root[110 #1^{2}+#1^{4}&, 4]) it is represented in. However, the common field found by ToNumberField contains the whole field ( Root[110 #1^{2}+#1^{4}&, 4]).
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Functions for computing algebraic number properties.
The minimal polynomial of an algebraic number a is the lowestdegree polynomial f with integer coefficients and the smallest positive leading coefficient, such that f (a)=0.
This gives the minimal polynomial of expressed as a pure function.
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This gives the minimal polynomial of Root[#1^{5}2 #1+7&, 1]^{2}+1 expressed as a polynomial in x.
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An algebraic number is an algebraic integer iff its MinimalPolynomial is monic.
This shows that is an algebraic integer.
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This shows that is not an algebraic integer.
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This gives the smallest positive integer n for which is an algebraic integer.
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The trace of an algebraic number a is the sum of all roots of MinimalPolynomial[a].
This gives the trace of (1)^{1/7}.
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The norm of an algebraic number a is the product of all roots of MinimalPolynomial[a].
An algebraic number a is an algebraic unit iff both a and 1/a are algebraic integers, or equivalently, iff AlgebraicNumberNorm[a] is 1 or 1.
An algebraic number a is a root of unity iff a^{n}=1 for some integer n.
This shows that is a root of unity.
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Functions for computing properties of elements of algebraic number fields.
If a is AlgebraicNumber[, coeffs], then MinimalPolynomial[a, x, Extension>Automatic] is equal to MinimalPolynomial[a, x]^{d}, where d is the extension degree of ()/ (a).
The trace of an algebraic number is the sum of all roots of its characteristic polynomial. If a is AlgebraicNumber[, coeffs], then AlgebraicNumberTrace[a, Extension>Automatic] is equal to d AlgebraicNumberTrace[a], where d is the extension degree of ()/ (a).
The norm of an algebraic number is the product of all roots of its characteristic polynomial. If a is AlgebraicNumber[, coeffs], then AlgebraicNumberNorm[a, Extension>Automatic] is equal to AlgebraicNumberNorm[a]^{d}, where d is the extension degree of ()/ (a).
Functions of computing properties of algebraic number fields.
An integral basis of an algebraic number field K is a list of algebraic numbers forming a basis of the module of the algebraic integers of K. The set {a_{1}, ..., a_{n}} is an integral basis of an algebraic number field K iff a_{i}K are algebraic integers, and every algebraic integer zK can be uniquely represented as
with integer coefficients k_{i}.
Here is an integral basis of (18^{1/3}).
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This gives an integral basis of the field generated by the first root of 533+429 #1+18 #1^{2}+#1^{3}&.
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This gives the roots of unity in the field generated by Root[92 #^{2}+#^{4}&,4].
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Here are all roots of unity in the field .
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{u_{1}, ..., u_{n}} is a list of fundamental units of an algebraic number field K iff u_{i}K are algebraic units, and every algebraic unit uK can be uniquely represented as
with a root of unity and integer exponents n_{i}.
Here is a set of fundamental units of the field generated by the third root of #1^{4}10 #1^{2}+1&.
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This gives a fundamental unit of the quadratic field .
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This gives a set of representatives of classes of elements of norm 9 in the field generated by the first root of #1^{2}7&.
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Here is a set of representatives of classes of elements of norm 2 in the field .
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This shows that the polynomial #^{5}+#^{4}+#^{3}+#^{2}+1& has 1 real root and 2 conjugate pairs of complex roots.
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This shows that the field [a] has 12 real embeddings and 6 conjugate pairs of complex embeddings.
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The discriminant of a number field K is the discriminant of an integral basis {a_{1}, ..., a_{n}} of K (i.e., the determinant of the matrix with elements AlgebraicNumberTrace[a_{i} a_{j}, Extension>Automatic]). The value of the determinant does not depend on the choice of integral basis.
This gives the discriminant of the field generated by a root of the polynomial #^{5}+#^{4}+#^{3}+#^{2}+1&. The value of the discriminant does not depend on the choice of the root; hence, NumberFieldDiscriminant allows specifying just the polynomial.
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The regulator of a number field K is the lattice volume of the image of the group of units of K under the logarithmic embedding
where _{1}, ..., _{s} are the real embeddings of K in , and _{s+1}, ..., _{s+t} are one of each conjugate pair of the complex embeddings of K in .
This gives the regulator of the field generated by a root of the polynomial #1^{3}3 #1^{2}+1&. The value of the regulator does not depend on the choice of the root; hence, NumberFieldRegulator allows specifying just the polynomial.
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