Splitting fields

This operation may be somewhat unusual and for specific interest. (Galois Group for example.) Procedurally, however, it is easy to obtain the splitting field of a polynomial by repeated application of algebraic factorization described in the previous section. The function is sp().

[103] sp(x^5-2);
[[x+(-#1),2*x+(#0^3*#1^3+#0^4*#1^2+2*#1+2*#0),2*x+(-#0^4*#1^2),2*x
+(-#0^3*#1^3),x+(-#0)],[[(#1),t#1^4+t#0*t#1^3+t#0^2*t#1^2+t#0^3*t#1+t#0^4],
[(#0),t#0^5-2]]]

Function sp() takes only one argument. The result is a list of two element: The first element is a list of linear factors, and the second one is a list whose elements are pairs (list of two elements) in the form [root, algptorat(defining polynomial)]. The second element, a list of pairs of form [root,algptorat(defining polynomial)], corresponds to the root's which are adjoined to eventually obtain the splitting field. They are listed in the reverse order of adjoining. Each of the defining polynomials in the list is, of course, guaranteed to be irreducible over the field obtained by adjoining all root's defined before it.

The first element of the result, a list of linear factors, contains all irreducible factors of the input polynomial over the field obtained by adjoining all root's in the second element of the result. Because such field is the splitting field of the input polynomial, factors in the result are all linear as the consequence.

Similarly to function af(), the product of all resulting factors may yield a polynomial which differs by a constant.

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