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KASH3 Reference Manual




Type of number fields. An object K of type fld^num is a finite field extension of a number field k or the rational numbers Q constructed as a quotient ring of the univariate polynomial ring over the base field modulo some irreducible polynomial: K = k[t]/(f(t)k[t]). Or, the field may be constructed as a multivariate quotient: K = k[s_1, ..., s_n]/(f_1(s_1), ..., f_n(s_n)) where all the polynomials are univariate. However, a slightly different representation is used internally. Note that Q is not a number field.
One has to distinguish between number fields with primitive element alpha:=K.1 which is a zero of f and number fields where no primitive element is known. In this case alpha_i:= K.i will be a zero of f_i.


x_QX:=PolynomialAlgebra(Q); x_f:=x_QX.1^2+2; x_F:=NumberField(x_f); Type(x_F);
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