3. Semilocalizations of the integers
Since in the following we need the semilocalizations Z_pi of the ring of integers (which are not already implemented as domains in the GAP library) as sources and ranges of rcwa mappings, we have to do this here.
3.1 Computing with semilocalizations of the integers
3.1-1 Z_pi
The ring Z_pi.
gap> R := Z_pi([2]);
Z_[ 2 ]
gap> S := Z_pi([2,5,7]);
Z_[ 2, 5, 7 ]
gap> T := Z_pi([3,11]);
Z_[ 3, 11 ]
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3.1-2 IsZ_pi
> IsZ_pi( R ) | ( property ) |
Indicates whether R is a ring Z_pi for some set of primes pi.
3.1-3 NoninvertiblePrimes
> NoninvertiblePrimes( R ) | ( attribute ) |
The noninvertible primes pi in the semilocalization R of the integers.
3.1-4 \in
Checks whether x is an element of the semilocalization R of the integers.
gap> 4/7 in R;
true
gap> 3/2 in R;
false
gap> 17/35 in S;
false
gap> 3/17 in S;
true
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3.1-5 Intersection
> Intersection( R, S ) | ( method ) |
The intersection of the semilocalizations R and S of the integers.
gap> U := Intersection(R,S,T);
Z_[ 2, 3, 5, 7, 11 ]
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3.1-6 IsSubset
> IsSubset( R, S ) | ( method ) |
Checks whether S is a subring of R, where R and S are semilocalizations of the integers.
gap> IsSubset(R,U);
true
gap> IsSubset(T,R);
false
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3.1-7 StandardAssociate
> StandardAssociate( R, x ) | ( method ) |
Returns the standard associate of x in the semilocalization R of the integers.
We define the standard associate of an element of Z_pi as the product of the noninvertible prime factors of its numerator.
gap> StandardAssociate(R,-6/7);
2
gap> StandardAssociate(R,36/5);
4
gap> StandardAssociate(U,37/13);
1
gap> StandardAssociate(U,36/13);
36
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3.1-8 GcdOp
> GcdOp( R, x, y ) | ( method ) |
Returns the greatest common divisor of x and y in the semilocalization R of the integers.
gap> GcdOp(S,-10/3,8/13);
2
gap> Gcd(S,90/3,60/17,120/33);
10
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3.1-9 LcmOp
> LcmOp( R, x, y ) | ( method ) |
Returns the least common multiple of x and y in the semilocalization R of the integers.
gap> LcmOp(S,-10/3,8/13);
40
gap> Lcm(S,90/3,60/17,120/33);
40
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3.1-10 Factors
> Factors( R, x ) | ( method ) |
Computes a prime factorization of x in the semilocalization R of the integers.
This method returns a list l, where the noninvertible prime factors of the numerator of x are stored in l{[2..Length(l)]} in ascending order, and l[1] is a suitable unit.
gap> Factors(U,840);
[ 2, 2, 2, 3, 5, 7 ]
gap> Factors(R,840);
[ 105, 2, 2, 2 ]
gap> Factors(R,-2/3);
[ -1/3, 2 ]
gap> Factors(S,60/17);
[ 3/17, 2, 2, 5 ]
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3.1-11 IsUnit
> IsUnit( R, x ) | ( method ) |
Checks whether x is a unit in the semilocalization R of the integers.
The method returns fail, if x is not an element of R.
gap> IsUnit(S,3/11);
true
gap> IsUnit(T,-2);
true
gap> IsUnit(T,0);
false
gap> IsUnit(T,3);
false
gap> IsUnit(T,3/11);
fail
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