In the lists below K usually refers to a number field, O to an order.
Number fields form the Magma category FldNum, orders form RngOrd. The notional power structures exist as parents of number fields and their orders, they allow no operations.
Given a number field K, this returns the number field over which K was defined by a single equation (so the last in a possible tower of subfields used in the definition of K). For an absolute number field K this will be the rational field Q.
Given a number field K, this returns an isomorphic number field L defined as an absolute extension (over Q).
Given number fields L and K such that Magma knows that K is a subfield of L, return an isomorphic number field M defined as an extension over K.
Given an order O, this returns the order over which O was defined by a single equation (so the last in a possible tower of orders used to define O). For an absolute order O this will be the integers Z.
Given an order O, this returns an isomorphic order O' defined as an order in an absolute extension (over Q).
Given an order O, obtained by a chain of transformations from an equation order E, return an order that is given directly by a single transformation over E.
Given an order O, return an order O' obtained from the old order by a transformation matrix T, which is returned as a second value, such that O' has a LLL-reduced basis.
Given an absolute order O, or an ideal I in an order, return the lattice determined by the real and complex embeddings of O or I.
This section contains two extremely useful functions for the
computation of all subfields or all subfields of a given degree
of an absolute number field.
Subfields(K, n) : FldNum -> [ < FldNum, Hom > ]
Given an absolute number field K and an integer greater than 1, this function returns a sequence of pairs (2-tuples) containing the subfields of K of degree n together with the embedding homomorphisms of each subfield into K. It is possible that the sequence contains isomorphic fields.
Given an absolute number field K, this function returns a sequence of pairs (2-tuples) containing the subfields of K (except Q) together with the embedding homomorphisms of each subfield into K. It is possible that the sequence contains isomorphic fields.
The lattice of subfields of an absolute number field K.
Given a number field K, return the degree [K:G] of K over its ground field G. For an order O it returns the relative degree of O over its ground order.
Given an order O or its field of fractions K, return the absolute degree of K over Q.
Given an absolute extension K of Q, return the discriminant of K. This discriminant is defined to be the discriminant of the maximal order of the field; as a consequence, its computation may trigger the determination of the maximal order. Note that the maximal order will be cached for possible future use.
The discriminant of any order defined over Z is by definition the discriminant of its basis, where the discriminant of any sequence of elements omega _i from K is defined to be the determinant of the trace matrix of the sequence.
The discriminant of absolute fields and orders is an integer.
Given an order O, return the discriminant of O. The discriminant in a relative extension is the ideal generated by the discriminant of all sequences of elements omega _i from O, where the discriminant of a sequence is defined to be the determinant of its trace matrix.
The discriminant of relative orders is an ideal of the base ring.
The reduced discriminant of the defining polynomial f of K, that is, the least positive integer in the ideal generated by f and its first derivative. Currently, this is only available for absolute extensions.This is the same as the maximal elementary divisor of the trace matrix of K.
Given a number field K, return the regulator of K, as a real number. (If the unit rank is 0, this returns 1.) Note that this will trigger the computation of the maximal order and its unit group if they are not known yet. This only works in an absolute extension.
Given an order O, return a lower bound on the regulator of O.
Given an absolute number field K, or an order O of K, return two integers, being the number of real embeddings and the number of pairs of complex embeddings of K.
The unit rank (one less than the number of real embeddings plus number of pairs of complex embeddings)
The index of order E in order O, for orders E subset O subset K.
Given a number field K, the polynomial (with rational coefficients) defining K as an extension of its ground field G is returned. For an order O a polynomial is returned defining O over its ground ring.
Given an absolute number field K or an order in K, and an integer n, return the zeroes of the defining polynomial of K with precision of exactly n decimal digits. The function returns a sequence of length the degree of K; all of the real zeroes will come before the complex ones in the sequence.
If n is omitted, the zeros are returned with the default precision.
Return the current basis for O over its ground ring as a sequence of elements of its field of fractions.
An integral basis for the absolute number field K is returned as a sequence of elements of K. This is the same as the basis for the maximal order. Note that the maximal order will be determined (and stored) if necessary.
Given an order O in a number field K of degree n, this returns a n x n matrix whose i-th row contains the (rational) coefficients for the i-th basis element of O with respect to the power basis of K. Thus, if b_i is the i-th basis element of O, b_i=sum_(j=1)^(n)M_(ij)alpha^(j - 1) where M is the matrix and alpha is the generator of K.The matrix is the same as TransformationMatrix(O, E), where E is the equation order for K, except that the coefficients are integers in this case.
Returns the transformation matrix for the transformation between the orders O and P in a common number field of degree n. The function returns a n x n matrix T with integer coefficients as well as a common integer denominator. The rows of the matrix express the n basis elements of O as a linear combination of the basis elements of P. Hence the effect of multiplying T on the left by a row vector v containing the basis coefficients of an element of O is the row vector v.T expressing the same element of the number field on the basis for P.
The functions Basis and IntegralBasis both return a sequence of elements, that can be accessed using the operators for enumerated sequences. Note that if, as in our example, O is the maximal order of K, both functions produce the same output:
> R<x> := PolynomialRing(Integers()); > f := x^4 - 420*x^2 + 40000; > K<y> := NumberField(f); > O := MaximalOrder(K); > I := IntegralBasis(K); > B := Basis(O); > I, B; [ (1), (1/2 * x), (1/4 * x + 1/40 * x^2 ), (1/2 + 9/40 * x + 1/800 * x^3 ) ] [ (1), (1/2 * x), (1/4 * x + 1/40 * x^2 ), (1/2 + 9/40 * x + 1/800 * x^3 ) ]The BasisMatrix function makes it possible to move between orders, in the following manner. We may regard orders as free Z-modules of rank the degree of the number field. The basis matrix then provides the transformation between the order and the equation order. The function ElementToSequence can be used to create module elements.
> K<x> := NumberField(x^4-420*x^2+40000); > O := MaximalOrder(K); > BM := BasisMatrix(O); > Mod := RSpace(RationalField(), Degree(K)); > z := O ! x; > e := z^2-3*z; > em := Mod ! ElementToSequence(e); > em; ( 0 -26 40 0) > f := em*BM; > f; ( 0 -3 1 0)So, since f is represented with respect to the basis of the equation order, which is the power basis, we indeed find the original element back. Of course it is much more useful to go in the other direction, using the inverse transformation (we check the result in the last line):
> E := EquationOrder(K); > f := y^3+7; > fm := Mod ! ElementToSequence(f); > e := fm*BM^-1; > e; (-393 -360 0 800) > &+[e[i]*B[i] : i in [1..Degree(K)] ]; x^3 + 7
Given an order O of some number field K of degree n, return the multiplication table with respect to its basis as a sequence of n matrices (with integer coefficients) of size n x n. The i-th matrix will have as its j-th row the basis representation of b_ib_j, where b_i is the i-th basis element for O.
Return the trace matrix of an order O, which has the trace Tr( omega _i x omega _j) as its i, j-th entry (where the omega _i are the basis for O).
The multiplication table of the order O consists of 4 matrices, such that the i-th 4 x 4 matrix (1 <= i <= 4) determines the multiplication by the i-th basis element of O (as a linear transformation with respect to that basis). Thus the third column of T[2] gives the basis coefficients for the product of B[2] and B[3], and we can use the sequence reduction operator to calculate B[2] * B[3] in an alternative way:
> R<x> := PolynomialRing(Integers());
> f := x^4 - 420*x^2 + 40000;
> K<y> := NumberField(f);
> O := MaximalOrder(K);
> B := Basis(O);
> B[2];
1/2*y
> T := MultiplicationTable(O);
> T[2];
[ 0 0 -5 -25]
[ 1 -5 -7 -7]
[ 0 10 5 15]
[ 0 0 10 0]
> &+[ T[2][i][3]*B[i] : i in [1..4] ];
1/80*y^3 + 1/8*y^2
> B[2]*B[3];
1/80*y^3 + 1/8*y^2
The trace matrix can either be found using the built-in function. or by
the one-line definition given below (for a field of degree 4):
> TraceMatrix(O); [ 4 0 21 2] [ 0 210 105 215] [ 21 105 173 118] [ 2 215 118 226] > MatrixRing(RationalField(), 4) ! [Trace(B[i]*B[j]): i, j in [1..4] ]; [ 4 0 21 2] [ 0 210 105 215] [ 21 105 173 118] [ 2 215 118 226]
Given two number fields K and L, this returns true as well as an isomorphism K -> L, if K and L are isomorphic, and it returns false otherwise.
Given two number fields K and L, this returns true as well as an embedding K hookrightarrow L, if K is a subfield of L, and it returns false otherwise.
Returns true if and only if the orders N and O, or the fields K and L, are the same.Two number fields will be the same if and only if their defining polynomials are the same; in fact the one structure will then necessarily be a copy of the other.
Two orders are equal if their fields of fraction are the same and the transformation matrix taking one to the other is integral and has determinant 1.
This returns true is the order O in the field K is the equation order of K, false otherwise.
This returns true if the order O in the field K is the maximal order of K, false otherwise.
This subsection describes the functions related to finding class groups and class numbers for (the maximal order O of) an absolute number field.
The method employed is the relation method, basically consisting of the following steps. In the first step a list of prime ideals of norm below a given bound is generated, the factor basis. In the second step a search is conducted to find in each of the prime ideals a few elements for which the principal ideal generated by it factors completely over the factor basis. Using these relations, a generating set for the ideal class group is derived (via matrix echelonization), and in the final step it is checked that the correct orders for the generators are found.
To determine the class group (or number) correctly one has to make sure that all ideals having norm smaller than the Minkowski bound (or smaller than the Bach bound if one assumes the generalized Riemann hypotheses) are taken into consideration, and that the final stage (which may be time consuming) is properly executed. Optional arguments allow the user to override these, but then correctness of the result can not be guaranteed.
Once a class group computation has been completed, the results are stored
with the number field.
ClassGroup(O: parameters) : RngOrd -> GrpAb, Map
Bound: RngIntElt Default: M
Euler: BoolElt Default: true
Verify: BoolElt Default: true
The group of ideal classes for the ring of integers O of the number field K is returned as an abelian group, together with a map from this abstract group to O.With the default values for the optional parameters the Minkowski bound is used and the last step of the algorithm verifies correctness (see the explanation above).
If Verify := false, the time-consuming verification step will not be executed. In this case the result is not guaranteed to be correct: it may be that the true order of the generators divides the one obtained.
To speed up the computation further, the user may instruct Magma to use a smaller bound than the Minkowski bound, by giving a positive integer value to Bound.
The parameter Euler is used to control testing of whether or not enough relations have been found: if its value is true (the default case) an Euler product computation is performed for this reason.
Bound: RngIntElt Default: M
Euler: BoolElt Default: true
Verify: BoolElt Default: true
Return the class number of the ring of integers O of a number field K. The meaning of the parameters is the same as for ClassGroup.
Bound: RngIntElt Default: M
Euler: BoolElt Default: true
Time: RngIntElt Default: t
Print: RngIntElt Default: l
Verify the correctness of the class group as found by the function ClassGroup. The function runs for t seconds: if it verifies correctness before t seconds then it returns the value { true}. Otherwise, it returns the value false. The latter means that it has failed to verify correctness within the stated time limit. In this situation no conclusions can be drawn from the return value.
Given an integral ideal I belonging to the maximal order of a number field, the ray class group modulo I is the quotient of the subgroup generated by the ideals coprime to I by the subgroup generated by the principal ideals generated by elements congruent to 1 modulo I.The sequence T contains the index numbers [ i_1, ..., i_r] of certain real infinite places. In this case, the generators of the principal ideals have to take positive values at the places indicated by the sequence T.
This function requires the class group to be known. If it is not already stored, it will be computed in such a way that its correctness is not guaranteed. (However, it will almost always be correct). If the user requires a guaranteed result, then the class group should be verified by the user.
The ray class group is returned as an abelian group, together with a bijection between this abstract group and a set of representatives for the ray classes.
Bound: RngIntElt Default: M
Euler: BoolElt Default: true
Verify: BoolElt Default: true
A sequence of integers, being the orders of cyclic factors of the class group of the ring of integers O of a number field K, is returned. For an explanation of the parameters, see the function ClassGroup above.
An (integer) upper bound for norms of generators of the ideal class group for K or O assuming the appropriate generalized Riemann hypotheses. If the second form is used, O must be a maximal order.
An unconditional (integer) upper bound for norms of the generators of the ideal class group for K. If the second form is used, O must be a maximal order.
Given the maximal order O, or a number field K with maximal order O, this function returns a sequence of prime ideals of norm less than a given bound B. This factor basis will not be stored with the order O, and does not affect possible class group and unit group computations.This bound is B, unless a factor basis for O with a different bound B' has been created before, in which case that is used. If no factor basis for O has been stored, one with bound B will be created, but it will not be stored with O, so that it is possible to create several factor bases with different bounds this way. Note however that factor bases used in the computation for unit groups and class groups will be stored with O.
Given a maximal order O, or a number field K with maximal order O, generate relations for each prime ideal in the factor basis for O with bound B on the norms of the ideals. The relations are given by columns in a matrix. If at some stage the relations generate the trivial group, no more relations are generated.
The relation method, outlined in the previous subsection, can also be used
for unit group calculations. Therefore
unit group calculations (including those triggered as a
side effect) may cause the creation of factor bases and relations.
There are some other methods (like Dirichlet's method) implemented, which
are faster under certain circumstances.
Descriptions can be found in [M. Pohst, H. Zassenhaus,
Algorithmic Algebraic Number Theory, Cambridge: Cambridge University
Press, 1989.], (pp. 343--344), and in [M. Pohst,
Computational Algebraic Number Theory, DMV Seminar Band 21,
Basel: Birkhäuser, 1993].
These methods will only work in the absolute case.
UnitGroup(O) : RngOrd -> GrpAb, Map
Al: MonStgElt Default: "Automatic"
Given an order O in a number field, this function returns an (abstract) abelian group U, as well as a bijection m between U and the units of the order. The unit group consists of the torsion subgroup, generated by the image m(U.1) and a free part, generated in O by the images m(U.i) for 2 <= i <= r_1 + r_2.If the argument to this function is a number field, the unit group of its maximal order is returned. Note that the maximal order may have to be determined first.
The parameter Al can be used to specify an algorithm. It should be one of "Automatic", (default, a choice will be made for the user) "ClassGroup", "Dirichlet", "Mixed" (the best known Dirichlet method), or "Relation". In the case of real quadratic fields, a continued fraction algorithm is available, ContFrac.
The torsion subgroup of the unit group of the order O, or, in case of a number field K, of its maximal order O. The torsion subgroup is returned as an abelian group, together with a map m from the group to the order O. The torsion subgroup will be cyclic, and is generated by m(T.1).
Al: MonStgElt Default: "Automatic"
Given an order O, this function returns a sequence of independent units; they generate a subgroup of finite index in the full unit group. Given a number field K it does the same for the maximal order of K. The function returns an abelian group (generated by the independent units) as well as a homomorphism from the group to the order.The parameter Al can be used to specify an algorithm. It should be one of "Automatic", (default, a choice will be made for the user) "ClassGroup", "Dirichlet", "Mixed" (the best known Dirichlet method), or "Relation". In the case of real quadratic fields, a continued fraction algorithm is available, ContFrac.
Given an order O in a number field, this function returns an (abstract) abelian group U, as well as a map m from U to the order. The image of U under m will be the p-part of the unit group of O, where p is a prime number. If a field K is given rather than an order, the above is computed for the maximal order of K.This function takes the same parameters as UnitGroup.
True iff the rank of the currently known unit group of O or K increases when a is merged into it.
Return the unit rank of the ring of integers O of a number field K.
> R<x> := PolynomialRing(Integers());
> f := x^4 - 420*x^2 + 40000;
> K<y> := NumberField(f);
> C := ClassGroup(K);
> C;
Abelian Group of order 1
> U := UnitGroup(K);
> U;
Abelian Group isomorphic to Z/2 + Z + Z + Z
Defined on 4 generators
Relations:
2*U.1 = 0
> T := TorsionUnitGroup(K);
> T;
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
2*$.1 = 0
An element x of an order O is an exceptional unit if both x and x - 1 are units in O. This function returns true iff u is an exceptional unit.
If u is an exceptional unit of an order O, then all of the units u_1=u, u_2=(1/u), u_3=1 - u, u_4=(1/(1 - u)), u_5=((u - 1)/u), u_6=(u/(u - 1)) are exceptional. The set Omega(u) formed by u_1, ..., u_6 is called the orbit of u. Usually it will have 6 elements. This function returns a sequence containing the elements of Omega(u).
This function returns a sequence S of units of O such that any exceptional unit u of O is either in S or is in the orbit of some element of S.
This function returns the group of s-units of the set of prime ideals given as either a sequence S or their product I, and the map from the group into the order the ideals lie in.[Next] [Prev] [Right] [Left] [Up] [Index] [Root]