Example FldAC_Cyclic6 (H43E5)

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);
30
> 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
}

Example FldAC_Split (H43E6)

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