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 asq(poly)

:: Squarefree factorization of polynomial poly over an
algebraic number field.
 af(poly,alglist)

 af_noalg(poly,defpolylist)

:: Factorization of polynomial poly over an
algebraic number field.
 return

list
 poly

polynomial
 alglist

root list
 defpolylist

root list of pairs of an indeterminate and a polynomial

Both defined in the file `sp'.

If the inputs contain no root's, these functions run fast
since they invoke functions over the integers.
In contrast to this, if the inputs contain root's, they sometimes
take a long time, since
cr_gcda() is invoked.

Function
af() requires the specification of base field,
i.e., list of root's for its second argument.

In the second argument
alglist , root defined last must come
first.

In
af(F,AL) , AL denotes a list of roots and it
represents an algebraic number field. In AL=[An,...,A1] each
Ak should be defined as a root of a defining polynomial
whose coefficients are in Q(A(k+1),...,An) .
[1] A1 = newalg(x^2+1);
[2] A2 = newalg(x^2+A1);
[3] A3 = newalg(x^2+A2*x+A1);
[4] af(x^2+A2*x+A1,[A2,A1]);
[[x^2+(#1)*x+(#0),1]]
To call sp_noalg , one should replace each algebraic number
ai in poly with an indeterminate vi. defpolylist
is a list [[vn,dn(vn,...,v1)],...,[v1,d(v1)]]. In this expression
di(vi,...,v1) is a defining polynomial of ai represented
as a multivariate polynomial.
[1] af_noalg(x^2+a2*x+a1,[[a2,a2^2+a1],[a1,a1^2+1]]);
[[x^2+a2*x+a1,1]]

The result is a list, as a result of usual factorization, whose elements
is of the form [factor, multiplicity].
In the result of
af_noalg , algebraic numbers in @v{factor} are
replaced by the indeterminates according to defpolylist.

The product of all factors with multiplicities counted may differ from
the input polynomial by a constant.
[98] A = newalg(t^22);
(#0)
[99] asq(x^4+6*x^3+(2*alg(0)9)*x^2+(6*alg(0))*x2);
[[x^2+3*x+(#0),2]]
[100] af(x^2+3*x+alg(0),[alg(0)]);
[[x+(#01),1],[x+(#0+2),1]]
[101] af_noalg(x^2+3*x+a,[[a,x^22]]);
[[x+a1,1],[x+a+2,1]]
 Reference

section
cr_gcda , section fctr , sqfr
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