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One may construct an absolute field isomorphic to the current
subfield represented by an algebraically closed field.
The construction of the absolute field may be very expensive,
as it involves factoring polynomials over successive subfields.
In fact, it is often the case that the degree of the absolute field is
an extremely large integer, so that an absolute field is not practically
representable, yet the system may allow one to compute effectively
with the original non-absolute presentation.
AbsoluteQuotientRing(A) : FldAC -> RngUPolRes
Simplify the algebraically closed field A fully (see Simplify(A) above) and then return an
absolute field as a univariate affine algebra R which is
isomorphic to the current (true) algebraic field represented by A, and
also return the isomorphism f:A -> R.
Simplify the algebraically closed field A fully (see Simplify(A) above) and then compute an
absolute field isomorphic to the current (true) algebraic field
represented by A and return the defining polynomial of
the absolute field. That is, return a polynomial f
such that K[x]/<f> is isomorphic to A in its current state.
Modify the algebraically closed field A in place so that has an absolute presentation.
That is, compute an isomorphic absolute field and absolute polynomial
f as in AbsolutePolynomial and modify A and its elements in
place so that A now only has one variable v and corresponding
defining polynomial f(v) and the elements of A correspond via
the isomorphism to their old representation.
We show how one can easily compute the number field over which the complete
variety of the Cyclic-6 ideal can be defined.
We first create the ideal I over Q and compute its variety over A,
the algebraic closure of Q.
> P<a,b,c,d,e,f> := PolynomialRing(RationalField(), 6);
> B := [
> a + b + c + d + e + f,
> a*b + b*c + c*d + d*e + e*f + f*a,
> a*b*c + b*c*d + c*d*e + d*e*f + e*f*a + f*a*b,
> a*b*c*d + b*c*d*e + c*d*e*f + d*e*f*a + e*f*a*b + f*a*b*c,
> a*b*c*d*e + b*c*d*e*f + c*d*e*f*a + d*e*f*a*b +
> e*f*a*b*c + f*a*b*c*d,
> a*b*c*d*e*f - 1];
> I := ideal<P | B>;
> time Groebner(I);
Time: 1.459
> A := AlgebraicClosure();
> time V := Variety(I, A);
Time: 4.219
> #V;
156
We now notice that there are 28 variables in A and we check that all
elements of V satisfy the original polynomials.
> Rank(A);
28
> V[1];
<-1, -1, -1, -1, r1 + 4, -r1>
> V[156];
<r28^3 + 2*r28^2*r9 - 2*r9, -r28^3 - 2*r28^2*r9 + 2*r9, r9, -r28, r28, -r9>
> {Evaluate(f, v): v in V, f in B};
{
0
}
We now simplify A to ensure that it represents a true field, and
prune away useless variables now having linear defining polynomials.
> time Simplify(A);
Time: 3.330
> Prune(A);
> A;
Algebraically closed field with 3 variables
Defining relations:
[
r3^2 - 1/3*r3*r2*r1 - 5/3*r3*r2 + 2/3*r3*r1 - 2/3*r3 + r2*r1 + 4*r2 + 1,
r2^2 - r2*r1 - 4*r1 - 1,
r1^2 + 4*r1 + 1
]
> V[1];
<-1, -1, -1, -1, r1 + 4, -r1>
> V[156];
<2/3*r3*r2*r1 + 7/3*r3*r2 - 1/3*r3*r1 - 2/3*r3 + 5/3*r2*r1 +
19/3*r2 + 2/3*r1 + 4/3, -2/3*r3*r2*r1 - 7/3*r3*r2 + 1/3*r3*r1
+ 2/3*r3 - 5/3*r2*r1 - 19/3*r2 - 2/3*r1 - 4/3, -4/3*r2*r1 -
14/3*r2 - 1/3*r1 - 2/3, 2/3*r3*r2*r1 + 7/3*r3*r2 - 1/3*r3*r1
- 2/3*r3 - 5/3*r2*r1 - 19/3*r2 + 1/3*r1 - 1/3, -2/3*r3*r2*r1
- 7/3*r3*r2 + 1/3*r3*r1 + 2/3*r3 + 5/3*r2*r1 + 19/3*r2 -
1/3*r1 + 1/3, 4/3*r2*r1 + 14/3*r2 + 1/3*r1 + 2/3>
Finally we compute an absolute polynomial for A, and then modify
A in place using Absolutize to make A be defined
by one polynomial of degree 8.
> time AbsolutePolynomial(A);
x^8 + 4*x^6 - 6*x^4 + 4*x^2 + 1
Time: 0.080
> time Absolutize(A);
Time: 0.259
> A;
Algebraically closed field with 1 variables
Defining relations:
[
r1^8 + 4*r1^6 - 6*r1^4 + 4*r1^2 + 1
]
> V[1];
<-1, -1, -1, -1, 1/2*r1^6 + 2*r1^4 - 7/2*r1^2 + 3, -1/2*r1^6 -
2*r1^4 + 7/2*r1^2 + 1>
> V[156];
<r1^7 + 4*r1^5 - 6*r1^3 + 4*r1, -r1^7 - 4*r1^5 + 6*r1^3 - 4*r1,
-1/4*r1^7 - 3/4*r1^5 + 11/4*r1^3 - 7/4*r1, -r1, r1, 1/4*r1^7
+ 3/4*r1^5 - 11/4*r1^3 + 7/4*r1>
> {Evaluate(f, v): v in V, f in B};
{
0
}
In this example we compute the splitting field of a certain polynomial
of degree 8.
We first set f to a degree-8 polynomial using the database of
polynomials with given Galois group. The Galois group has order
16, so we know that the splitting field will have absolute degree
16.
> P<x> := PolynomialRing(IntegerRing());
> load galpols;
Loading "/home/magma/libs/galpols/galpols"
> PolynomialWithGaloisGroup(8, 6);
x^8 - 2*x^7 - 9*x^6 + 10*x^5 + 22*x^4 - 14*x^3 - 15*x^2 + 2*x + 1
> f := $1;
> #GaloisGroup(f);
16
We next create an algebraic closure A and compute the roots of f
over A.
> A := AlgebraicClosure();
> r := Roots(f, A);
> #r;
8
> A;
Algebraically closed field with 8 variables
Defining relations:
[
r8^8 - 2*r8^7 - 9*r8^6 + 10*r8^5 + 22*r8^4 - 14*r8^3 - 15*r8^2 + 2*r8 + 1,
r7^8 - 2*r7^7 - 9*r7^6 + 10*r7^5 + 22*r7^4 - 14*r7^3 - 15*r7^2 + 2*r7 + 1,
r6^8 - 2*r6^7 - 9*r6^6 + 10*r6^5 + 22*r6^4 - 14*r6^3 - 15*r6^2 + 2*r6 + 1,
r5^8 - 2*r5^7 - 9*r5^6 + 10*r5^5 + 22*r5^4 - 14*r5^3 - 15*r5^2 + 2*r5 + 1,
r4^8 - 2*r4^7 - 9*r4^6 + 10*r4^5 + 22*r4^4 - 14*r4^3 - 15*r4^2 + 2*r4 + 1,
r3^8 - 2*r3^7 - 9*r3^6 + 10*r3^5 + 22*r3^4 - 14*r3^3 - 15*r3^2 + 2*r3 + 1,
r2^8 - 2*r2^7 - 9*r2^6 + 10*r2^5 + 22*r2^4 - 14*r2^3 - 15*r2^2 + 2*r2 + 1,
r1^8 - 2*r1^7 - 9*r1^6 + 10*r1^5 + 22*r1^4 - 14*r1^3 - 15*r1^2 + 2*r1 + 1
]
Finally we simplify A. There are defining polynomials of degrees 2 and
8 in the simplified field. The absolute polynomial of degree 16
defines the splitting field of f.
> time Simplify(A);
Time: 2.870
> A;
Algebraically closed field with 8 variables
Defining relations:
[
r8 + 1/2*r3*r1^6 - 2*r3*r1^5 - r3*r1^4 + 8*r3*r1^3 - 2*r3*r1^2 -
5*r3*r1 - 1/2*r3 + r1^7 - 3/2*r1^6 - 9*r1^5 + 4*r1^4 + 19*r1^3
- r1^2 - 9*r1 - 5/2,
r7 - 1/2*r3*r1^6 + 2*r3*r1^5 + r3*r1^4 - 8*r3*r1^3 + 2*r3*r1^2 +
5*r3*r1 + 1/2*r3,
r6 + r3 - 3/2*r1^7 + 2*r1^6 + 14*r1^5 - 5*r1^4 - 28*r1^3 + 3*r1^2
+ 19/2*r1,
r5 - 1/2*r3*r1^6 + 2*r3*r1^5 + r3*r1^4 - 8*r3*r1^3 + 2*r3*r1^2 +
4*r3*r1 + 1/2*r3 + r1^6 - r1^5 - 9*r1^4 - r1^3 + 14*r1^2 +
6*r1,
r4 + 1/2*r3*r1^6 - 2*r3*r1^5 - r3*r1^4 + 8*r3*r1^3 - 2*r3*r1^2 -
4*r3*r1 - 1/2*r3 - 1,
r3^2 - 3/2*r3*r1^7 + 2*r3*r1^6 + 14*r3*r1^5 - 5*r3*r1^4 -
28*r3*r1^3 + 3*r3*r1^2 + 19/2*r3*r1 + 3/2*r1^6 - r1^5 -
15*r1^4 - 4*r1^3 + 27*r1^2 + 11*r1 - 9/2,
r2 + 1/2*r1^7 - 3/2*r1^6 - 4*r1^5 + 10*r1^4 + 10*r1^3 - 16*r1^2 -
11/2*r1 + 3/2,
r1^8 - 2*r1^7 - 9*r1^6 + 10*r1^5 + 22*r1^4 - 14*r1^3 - 15*r1^2 +
2*r1 + 1
]
> AbsolutePolynomial(A);
x^16 - 36*x^14 + 488*x^12 - 3186*x^10 + 10920*x^8 - 19804*x^6 + 17801*x^4 -
6264*x^2 + 64
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