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# Edit detail for SandBoxPolynomialOverFiniteField revision 3 of 3

 1 2 3 Editor: hemmecke Time: 2014/11/25 07:42:13 GMT+0 Note:

added:
In order to work with a finite field of order $p^n$ where $p$ is a prime, define
the second argument of UnivariatePolynomial as FiniteField(p, n).

\begin{axiom}
F ==> FiniteField(p, 2)
b: Vector F := basis()$F t: F := b.2 1+t+t^2+t^3 PF ==> UnivariatePolynomial('y, F) y: PF := monomial(1,1)$PF
pf: PF := (y+2)*(2*y^5 + y^3 + 2)
pg: PF := (2*t+1)*y^3 + (1+t)*y^2 -t*y +1
pf/pg
factor pf
factor pg
\end{axiom}

In case the field property is not needed, IntegerMod would be the way to go.
But note that in general there are zero divisors, so division will not work
(even not in the case where the argument of IntegerMod is a prime).
The reason is IntegerMod doesn't check whether it's argument is prime and so
does not know that it might be an integral domain.

\begin{axiom}
R ==> IntegerMod(p)
PR ==> UnivariatePolynomial('z, R)
z: PR := monomial(1,1)$PR rf: PR := (z+2)*(2*z^5 + z^3 + 2) rg: PR := 2*z^3 + z^2 -z +1 rf/rg F has IntegralDomain factor rf factor rg \end{axiom} Also note that factorization of rf fails. The reason most probably is that the used algorithm needs to divide by some coefficient and it fails to recognize that this is in fact possible (at least for primes p). In IntegerMod, there simply is no division. \begin{axiom} r1: R := 2 r2: R := 1 r1/r2 r2/r1 \end{axiom}  fricas p ==> 3 Type: Void fricas K ==> PrimeField p Type: Void fricas P ==> UnivariatePolynomial('x, K) Type: Void fricas x: P := monomial(1,1)$P
 (1)
Type: UnivariatePolynomial(x,PrimeField?(3))
fricas
f: P := (x+2)*(2*x^5 + x^3 + 2)
 (2)
Type: UnivariatePolynomial(x,PrimeField?(3))
fricas
g: P := 2*x^3 + x^2 -x +1
 (3)
Type: UnivariatePolynomial(x,PrimeField?(3))
fricas
f/g
 (4)
Type: Fraction(UnivariatePolynomial(x,PrimeField?(3)))
fricas
factor f
 (5)
Type: Factored(UnivariatePolynomial(x,PrimeField?(3)))
fricas
factor g
 (6)
Type: Factored(UnivariatePolynomial(x,PrimeField?(3)))
fricas
k: K := 2
 (7)
Type: PrimeField?(3)
fricas
f/k
 (8)
Type: UnivariatePolynomial(x,PrimeField?(3))

In order to work with a finite field of order where is a prime, define the second argument of UnivariatePolynomial as FiniteField?(p, n).

fricas
F ==> FiniteField(p, 2)
Type: Void
fricas
b: Vector F := basis()$F  (9) Type: Vector(FiniteField?(3,2)) fricas t: F := b.2  (10) Type: FiniteField?(3,2) fricas 1+t+t^2+t^3  (11) Type: FiniteField?(3,2) fricas PF ==> UnivariatePolynomial('y, F) Type: Void fricas y: PF := monomial(1,1)$PF
 (12)
Type: UnivariatePolynomial(y,FiniteField?(3,2))
fricas
pf: PF := (y+2)*(2*y^5 + y^3 + 2)
 (13)
Type: UnivariatePolynomial(y,FiniteField?(3,2))
fricas
pg: PF := (2*t+1)*y^3 + (1+t)*y^2 -t*y +1
 (14)
Type: UnivariatePolynomial(y,FiniteField?(3,2))
fricas
pf/pg
 (15)
Type: Fraction(UnivariatePolynomial(y,FiniteField?(3,2)))
fricas
factor pf
 (16)
Type: Factored(UnivariatePolynomial(y,FiniteField?(3,2)))
fricas
factor pg
 (17)
Type: Factored(UnivariatePolynomial(y,FiniteField?(3,2)))

In case the field property is not needed, IntegerMod? would be the way to go. But note that in general there are zero divisors, so division will not work (even not in the case where the argument of IntegerMod? is a prime). The reason is IntegerMod? doesn't check whether it's argument is prime and so does not know that it might be an integral domain.

fricas
R ==> IntegerMod(p)
Type: Void
fricas
PR ==> UnivariatePolynomial('z, R)
Type: Void
fricas
z: PR := monomial(1,1)$PR  (18) Type: UnivariatePolynomial(z,IntegerMod?(3)) fricas rf: PR := (z+2)*(2*z^5 + z^3 + 2)  (19) Type: UnivariatePolynomial(z,IntegerMod?(3)) fricas rg: PR := 2*z^3 + z^2 -z +1  (20) Type: UnivariatePolynomial(z,IntegerMod?(3)) fricas rf/rg There are 12 exposed and 12 unexposed library operations named / having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op / to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named / with argument type(s) UnivariatePolynomial(z,IntegerMod(3)) UnivariatePolynomial(z,IntegerMod(3)) Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.
F has IntegralDomain
 (21)
Type: Boolean
fricas
factor rf
 (22)
Type: Factored(UnivariatePolynomial(z,Integer))
fricas
factor rg
 (23)
Type: Factored(UnivariatePolynomial(z,Integer))

Also note that factorization of rf fails. The reason most probably is that the used algorithm needs to divide by some coefficient and it fails to recognize that this is in fact possible (at least for primes p). In IntegerMod?, there simply is no division.

fricas
r1: R := 2
 (24)
Type: IntegerMod?(3)
fricas
r2: R := 1
 (25)
Type: IntegerMod?(3)
fricas
r1/r2
There are 12 exposed and 12 unexposed library operations named /
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op /
to learn more about the available operations. Perhaps
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named /
with argument type(s)
IntegerMod(3)
IntegerMod(3)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need. r2/r1 There are 12 exposed and 12 unexposed library operations named / having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op / to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named / with argument type(s) IntegerMod(3) IntegerMod(3) Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.