GAP Manual: 16.18. DecomPoly

DecomPoly( pol )
DecomPoly( pol, "all" )

returns an ideal decomposition of the polynomial pol. An ideal decomposition is given by two polynomials g and h, such that pol divides (gcirch). By the Galois correspondence any ideal decomposition corresponds to a block system of the Galois group. The polynomial g defines a subfield K(beta) of K(alpha) with h(alpha)=beta. The first form finds one ideal decomposition, while the second form finds all possible different ideal decompositions (i.e. all subfields).

    gap> d:=DecomPoly(e.minpol);
    [ ]
    gap> p:=x^10-10*x^9+20*x^8+25*x^7+(-1885/8)*x^6+(109/4)*x^5
    > +(3635/8)*x^4+(4535/16)*x^3+(-456615/256)*x^2+(135315/128)*x
    > +(-64681/128);;
    gap> d:=DecomPoly(p,"all");
    [ [ x^5 - 608253837790304032960320*x^4 + 
	    102513277599629696329909384806667776261204935040*x^3 - 
	    549049420457453617648141380902484571131598386028760933387027\
    6218920960*x^2 + 
	    286387493717897513685666250304380673273734943195175651689786\
    74636841864755780846410697052160*x - 
	    173134960201194027097404571687818402186256514770030882717264\
    9557460578764008149662018624414041261233616423188430848, 
          8772892716132663808*x^9 - 52775093681915390720*x^8 - 
            205982735485386544256*x^7 + 1328709758167472769984*x^6 - 
            2621982207697769174432*x^5 - 7339219574720832743376*x^4 + 
            9552106782081018334600*x^3 + 15478181913028444442212*x^2 - 
            46541684909437606849432*x + 16156582976797641690520 ] ]
    gap> Degree(Value(d[1][1],d[1][2])/f);
    35