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KASH3 Reference Manual


Orders of Number Fields


KASH3 makes it easy to compute in arbitrary orders of number fields. Given the minimal polynomial 'f' of an algebraic integer 'rho' one obtains the equation order 'Z[rho]' easily as 'Z[x]/(rho)'. In order to compute a maximal order 'O' of the number field 'F=Q(rho)', one has to compute an integral bases of 'F'. Maximal orders are not given by polynomials but a transformation matrix, which transforms a power basis '(1,rho,...,rho^4)' to a basis '(w_1,...,w_5)' of the maximal order. The 'MaximalOrder' function computes an integral basis '(w_1,...,w_5)'. Using the 'Element' function one can enter algebraic numbers with respect to this basis.


f := X^5 + 4*X^4 - 56*X^2 -16*X + 192; o := EquationOrder(f); O := MaximalOrder(o); # maximal order of 'o' w1 := Element(O,[1,0,0,0,0]); w2 := Element(O,[0,1,0,0,0]); w3 := Element(O,[0,0,1,0,0]); w4 := Element(O,[0,0,0,1,0]); w5 := Element(O,[0,0,0,0,1]);
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