The field \(\mathbb{Q}^{\sqrt{}}(\imath)\) with \(\mathbb{Q}^{\sqrt{}}=\mathbb{Q}(\{\sqrt{p}\mid p\textrm{ a prime}\})\) and \(\imath=\sqrt{-1}\in\mathbb{C}\) is realised as SqrtField. A few functions print some information on what they are doing to the info class InfoSqrtField; this can be turned off by setting SetInfoLevel( InfoSqrtField, 0 );.
‣ SqrtFieldIsGaussRat( q ) | ( function ) |
Here q is an element of SqrtField; this function returns true if and only if q is the product of One(SqrtField) and a Gaussian rational.
gap> F := SqrtField; SqrtField gap> IsField( F ); LeftActingDomain( F ); Size( F ); Characteristic( F ); true GaussianRationals infinity 0 gap> one := One( F ); 1 gap> 2 in F; 2*one in F; 2*E(4)*one in F; false true true gap> a := 2/3*E(4)*one;; gap> a in SqrtField; a in GaussianRationals; SqrtFieldIsGaussRat( a ); true false true
Every \(f\) in SqrtField can be uniquely written as \(f=\sum_{j=1}^m r_i \sqrt{k_j}\) for Gaussian rationals \(r_i\in\mathbb{Q}(\imath)\) and pairwise distinct squarefree positive integers \(k_1,\ldots,k_m\). Thus, \(f\) can be described efficiently by its coefficient vector \([[r_1,k_1],\ldots,[r_j,k_j]]\).
‣ Sqroot( q ) | ( function ) |
Here q is a rational number and Sqroot(q) is the element \(\sqrt{q}\) as an element of SqrtField. If \(q=(-1)^\epsilon a/b\) with coprime integers \( a,b\geq 0\) and \(\epsilon\in\{0,1\}\), then Sqroot(q) is represented as the element E(4)\(^\varepsilon\)*b*Sqroot(ab) of SqrtField.
‣ CoefficientsOfSqrtFieldElt( f ) | ( function ) |
If f is an element in SqrtField, then CoefficientsOfSqrtFieldElt(f) returns its coefficient vector \([[r_1,k_1],\ldots,[r_m,k_m]]\) as described above, that is, \(r_1,\ldots,r_m\in\mathbb{Q}(\imath)\) and \(k_1,\ldots,k_m\) are pairwise distinct positive squarefree integers such that \(f=\sum_{j=1}^m r_j\sqrt{k_j}\).
‣ SqrtFieldEltByCoefficients( l ) | ( function ) |
If l is a list \([[r_1,k_1],\ldots,[r_m,k_m]]\) with Gaussian rationals \(r_j\) and rationals \(k_j\), then SqrtFieldEltByCoeffiients(l) returns the element \(\sum_{j=1}^m r_j\sqrt{k_j}\) as an element of SqrtField. Note that here \(k_1,\ldots,k_m\) need not to be positive, squarefree, or pairwise distinct.
gap> Sqroot(-(2*3*4)/(11*13)); Sqroot(245/15); Sqroot(16/9); 2/143*E(4)*Sqroot(858) 7/3*Sqroot(3) 4/3 gap> a := 2+Sqroot(7)+Sqroot(99); 2 + Sqroot(7) + 3*Sqroot(11) gap> CoefficientsOfSqrtFieldElt(a); [ [ 2, 1 ], [ 1, 7 ], [ 3, 11 ] ] gap> SqrtFieldEltByCoefficients([[2,9],[1,7],[E(4),13]]); 6 + Sqroot(7) + E(4)*Sqroot(13)
‣ SqrtFieldEltToCyclotomic( f ) | ( function ) |
If f lies in SqrtField with coefficient vector \([[r_1,k_1],\ldots,[r_m,k_m]]\), then SqrtFieldEltToCyclotomic(f) returns \(\sum_{j=1}^m r_j\sqrt{k_j}\) lying in a suitable cyclotomic field CF(n). The degree \(n\) can easily become too large, hence this function should be used with caution.
‣ SqrtFieldEltByCyclotomic( c ) | ( function ) |
If c is an element of \(\mathbb{Q}^{\sqrt{}}(\imath)\) represented as an element of a cyclotomic field CF(n), then SqrtFieldEltByCyclotomic(c) returns the corresponding element in SqrtField. Our algorithm for doing this is described in [DG13].
gap> SqrtFieldEltToCyclotomic( Sqroot(2) ); E(8)-E(8)^3 gap> SqrtFieldEltToCyclotomic( Sqroot(2)+E(4)*Sqroot(7) ); E(56)^5+E(56)^8+E(56)^13-E(56)^15+E(56)^16-E(56)^23-E(56)^24+E(56)^29-E(56)^31 +E(56)^32+E(56)^37-E(56)^39-E(56)^40+E(56)^45-E(56)^47-E(56)^48+E(56)^53 -E(56)^55 gap> SqrtFieldEltByCyclotomic( E(8)-E(8)^3 ); Sqroot(2) gap> SqrtFieldEltByCyclotomic( 3*E(4)*Sqrt(11)-2/4*Sqrt(-13/7) ); 3*E(4)*Sqroot(11) + (-1/14*E(4))*Sqroot(91)
All basic field operations are available. The inverse of an element \(f\) in SqrtField as follows: We first compute the minimal polynomial \(p(X)\) of \(f\) over \(\mathbb{Q}(\imath)\), that is, a non-trivial linear combination \(0=p(f)=a_0+a_1 f+\ldots a_{i-1}f^{i-1}+f^i\). Then \(f^{-1}=-(a_1+a_2f+\ldots+a_{i-1}f^{i-2}+f^{i-1})/a_0\). Although the inverse of \(f\) can be computed with linear algebra methods only, the degree of the minimal polynomial of \(f\) can become rather large. For example, if \(f=\sum_{j=1}^m r_i \sqrt{k_j}\) for rational \(r_i\) and pairwise distinct positive squarefree integers \(k_1,\ldots,k_m\), then \(f\) is a primitive element of the number field \(\mathbb{Q}(\sqrt{k_1},\ldots,\sqrt{k_m})\), see for example Lemma A.5 in [DG13]. For larger degree, the progress of the computation of the inverse is printed via the InfoClass InfoSqrtField. We remark that the method Random simply returns a sum of a few terms \(a\sqrt{b}\) where \(a,b\) are random rationals constructed with Random(Rationals).
gap> a := Sqroot( 2 ) + 3 * Sqroot( 3/7 ); b := Sqroot( 21 ) - Sqroot( 2 ); Sqroot(2) + 3/7*Sqroot(21) (-1)*Sqroot(2) + Sqroot(21) gap> a + b; a * b; a - b; 10/7*Sqroot(21) 7 + 4/7*Sqroot(42) 2*Sqroot(2) + (-4/7)*Sqroot(21) gap> c := ( a - b )^-2; 91/8 + 7/4*Sqroot(42) gap> a := Sum( List( [2,3,5,7], x -> Sqroot( x ) ) ); Sqroot(2) + Sqroot(3) + Sqroot(5) + Sqroot(7) gap> b := a^-1; a*b; 37/43*Sqroot(2) + (-29/43)*Sqroot(3) + (-133/215)*Sqroot(5) + 27/43*Sqroot( 7) + 62/215*Sqroot(30) + (-10/43)*Sqroot(42) + (-34/215)*Sqroot(70) + 22/ 215*Sqroot(105) 1 gap> ComplexConjugate(Sqroot(17)+Sqroot(-7)); (-E(4))*Sqroot(7) + Sqroot(17) gap> Random( SqrtField ); E(4) + (-7/6+1/4*E(4))*Sqroot(2) + (-3/2*E(4))*Sqroot(3)
Most methods for list, matrices, and polynomials also work over SqrtField.
gap> m:=[[Sqroot(2),Sqroot(3)],[Sqroot(2),Sqroot(5)],[1,0]]*One(SqrtField); [ [ Sqroot(2), Sqroot(3) ], [ Sqroot(2), Sqroot(5) ], [ 1, 0 ] ] gap> NullspaceMat(m); [ [ (-5/4)*Sqroot(2) + (-1/4)*Sqroot(30), 3/4*Sqroot(2) + 1/4*Sqroot(30), 1 ] ] gap> RankMat(m); 2 gap> m := [[Sqroot(2),Sqroot(3)],[Sqroot(2),Sqroot(5)]]; [ [ Sqroot(2), Sqroot(3) ], [ Sqroot(2), Sqroot(5) ] ] gap> Determinant( m ); DefaultFieldOfMatrix( m ); (-1)*Sqroot(6) + Sqroot(10) SqrtField gap> x := Indeterminate( SqrtField, "x" );; f := x^2+x+1; x^2+x+1
‣ SqrtFieldMakeRational( m ) | ( function ) |
If m is an element of SqrtField, or a list or a matrix over SqrtField, defined over the Gaussian rationals, then SqrtFieldMakeRational( m ) returns the corresponding element in \(\mathbb{Q}(\imath)\) or defined over \(\mathbb{Q}(\imath)\), respectively. This function is used internally, for example, to compute the determinant or rank of a rational matrix over SqrtField more efficiently. It is also used in the following three functions.
‣ SqrtFieldPolynomialToRationalPolynomial( f ) | ( function ) |
Here f is a polynomial over SqrtField but with coefficients in the Gaussian rationals. The function returns the corresponding polynomial defined over the Gaussian rationals.
‣ SqrtFieldRationalPolynomialToSqrtFieldPolynomial( f ) | ( function ) |
If f is a polynomial over the Gaussian rationals, then the function returns the corresponding polynomial defined over SqrtField.
‣ Factors( f ) | ( operation ) |
If f is a rational polynomial defined over SqrtField, then the previous two functions are used to obtain its factorisation over \(\mathbb{Q}\).
gap> F := SqrtField;; one := One( SqrtField );; gap> x := Indeterminate( F, "x" );; f := x^5 + 4*x^3 + E(4)*one*x; x^5+4*x^3+E(4)*x gap> SqrtFieldPolynomialToRationalPolynomial(f); x_1^5+4*x_1^3+E(4)*x_1 gap> SqrtFieldRationalPolynomialToSqrtFieldPolynomial(last); x^5+4*x^3+E(4)*x gap> f := x^2-1;; Factors(f); [ x-1, x+1 ] gap> f := x^2+1;; Factors(f); [ x^2+1 ]
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