Factored
Objects kept in factored form
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Factored(list
, <type
>, <ring
>) Factored(f
, <type
>, <ring
>)
Factored
is the domain of objects kept in
factored form, such as prime factorization of integers, squarefree
factorization of polynomials, or the factorization of polynomials
in irreducible factors.
The argument list
must be a list of odd length
and of the form [u, f1, e1, f2, e2, ..., fr, er]
,
where the entries u and f_{i} are
elements of the domain ring
, or can be converted
into such elements. The e_{i} must
be integers. Here, i ranges
from 1 to r.
See section “Operands” below for the meaning of the entries of that list.
An error message is reported, if one of the list entries is of wrong type.
An arithmetical expression f
given as the
first argument is the same as giving the list [ring::one,
f, 1]
.
See section “Operands” below for the meaning of the entries of that list.
f
must be an element of the domain ring
,
or must be convertible into such an element, otherwise an error message
would be given.
The argument type
indicates what is known
about the factorization. Currently, the following types are known:
"unknown"
– nothing is known
about the factorization.
"irreducible"
– the f_{i} are
irreducible over the domain ring
.
"squarefree"
– the f_{i} are
squarefree over the domain ring
.
If this argument is missing, then the type of the created factored
object is set to "unknown"
.
The type of factorization is known to any element of Factored
.
Use the methods "getType"
and "setType"
(see
below) to read and set the type of factorization of a given factored
object.
The argument ring
is the ring of factorization.
It must be an integral domain, i.e., a domain of category Cat::IntegralDomain
.
If this argument is missing, then the domain Dom::ExpressionField()
is
used.
The ring of factorization is known to any element of Factored
.
Use the methods "getRing"
and "setRing"
(see
below) to read and set the ring of factorization of a given factored
object.
You can use the index operator [ ]
to extract the factors
of an element f
of the domain Factored
.
E.g., for f = u f_{1}^{e1} f_{2}^{e2} …,
you have f [1] = u, f [2]
= f_{1}^{e1}, f [3]
= f_{2}^{e2} etc.
You can also use the methods "factors"
and "exponents"
(see
below) to access the operands, i.e., the call Factored::factors(f)
returns
a list of the factors f_{i},
and Factored::exponents(g)
returns a list of the
exponents e_{i} (1
≤ i ≤ r).
The system functions ifactor
, factor
and polylib::sqrfree
are
the main application of this domain, they return their result in form
of such factored objects (see their help pages for information about
the type and ring of factorization).
There may be no need to explicitly create factored objects, but to work with the results of the mentioned system functions.
Note that an element of Factored
is printed
like an expression and behaves like that. As an example, the result
of f := factor(x^2 + 2*x + 1)
is an element of Factored
and
printed as (x + 1)^2
. The call type(f)
returns "_power"
as
the expression type of f
.
For an element f
of Factored
,
the call Factored::convert(f, DOM_LIST)
gives a
list of all operands of f
.
The following computes the prime factorization of the integer 20:
f := ifactor(20)
The result is an element of the domain Factored
:
domtype(f)
which consists of the following five operands:
op(f)
They represent the integer 20 in
the following form: 20 = 1 2^{2} 5.
The factors are prime numbers and can be extracted via Factor::factors
:
Factored::factors(f)
ifactor
kept
the information that the factorization ring is the ring of integers
(represented by the domain Dom::Integer
),
and that the factors of f
are prime (and therefore
irreducible, because ℤ is
an integral domain):
Factored::getRing(f), Factored::getType(f)
We can convert such an object into different forms, such as into a list of its operands:
Factored::convert_to(f, DOM_LIST)
or into an unevaluated expression, keeping the factored form:
Factored::convert_to(f, DOM_EXPR)
or back into an integer:
Factored::convert_to(f, Dom::Integer)
You may also use the system function coerce
here, which has the same effect.
We compute the factorization of the integers 108 and 512:
n1 := ifactor(108); n2 := ifactor(512)
The multiplication of these two integers gives the prime factorization of 55296 = 108 512:
n1*n2
Note that the most operations on such objects lead to an unfactored form, such as adding these two integers:
n1 + n2
You may apply the function ifactor
to the result, if you are interested
in its prime factorization:
ifactor(%)
You an apply (almost) each function to factored objects, functions that mainly expect arithmetical expressions as their input. Note that, before the operation is applied, the factored object is converted into an arithmetical expression in unfactored form:
Re(n1)
The second system function which deals with elements of Factored
,
is factor
, which
computes all irreducible factors of a polynomial.
For example, if we define the following polynomial of ℤ_{101}:
p := poly(x^12 + x + 1, [x], Dom::IntegerMod(101)):
and compute its factorization into irreducible factors, we get:
f := factor(p)
If we multiply the factored object with an element that can
be converted into an element of the ring of factorization, then we
get a new factored object, which then is of the factorization type "unknown"
:
x*f
Factored::getType(%)
You may use the function expand
which returns the factored object
in expanded form as an element of the factorization ring:
expand(f)
The third system function which return elements of Factored
is polylib::sqrfree
, which
computes the squarefree factorization of polynomials. For example:
f := polylib::sqrfree(x^2 + 2*x + 1)
The factorization type, of course, is "squarefree"
:
Factored::getType(f)

A list of odd length 

An arithmetical expression 

A string (default: 

A domain of category 
Calling a factored object as a function yields the object itself, regardless of the arguments. The arguments are not evaluated.
You can apply (almost) every function to factored objects, functions that mainly expect arithmetical expressions as their input.
For example, one may add or multiply those objects, or apply
functions such as expand
and diff
to them. But the
result of such an operation then is usually not any longer of the
domain Factored
, as the factored form could be
lost due to the operation (see examples below).
Call expr(f)
to convert the factored object f
into
an arithmetical expression (as an element of a kernel domain).
The call coerce(f, DOM_LIST)
returns a list
of operands of the factored object f
(see method "convert_to"
below).
An element f of Factored
consists
of the r + 1 operands u, f_{1}, e_{1}, f_{2}, e_{2},
…, f_{r}, e_{r},
such that f = u f_{1}^{e1} f_{2}^{e2} … f_{r}^{er}.
The first operand u and
the factors f_{i} are
elements of the domain ring
. The exponents e_{i} are integers.