e1 = lfirst(L2);		L2 = lred(L2);     		z  = lfirst(L);		z  = lfirst(z);		e1 = cdprfmsp1mh(p,e1,z);		if ( e1 != 0 ) {			G = cdprfmsp1lfm(L,p);			do {				Gl = lfirst(G);				G  = lred(G);				h  = afmsp1prodsp(p,A1,e1,Gl,list2(0,1));				if ( h != 0 ) {					Ml = clfcdprfmsp1(h,n1);					M  = lcomp(Ml,M);				}                	} while (G!=_0);		}   	}	M = cdmarfmsp1hr(p,M);	return(M);} static list coreal(p,F,P,Q,mp,neu,zaehl)single p;              pol F,P,Q,mp;single neu,zaehl;{	single n,na,wy,e4,r,i,e,k,l,d,t,ph,ph2;	int  ph3;	rat  va,vb,vc;	list Vax,H1,H2,H3,H4,HH,M,B,B2,Bl,AL;	pol  Fh,Pp,P2,Ph,MOD,R2,E,U,G,H,y;	pol  a0,ma0,a1,a2,alb,b0,mb0,b1,b2,blb,c0,mc0,c1,c2,clb;	bind(F,P,Q,mp);	init(ph3,va,vb,vc,Vax,H1,H2,H3,H4,HH,M,B,B2,Bl,AL,Fh,Pp,P2,Ph,MOD,R2,E,U,G,H,y);	init(a0,ma0,a1,a2,alb,b0,mb0,b1,b2,blb,c0,mc0,c1,c2,clb);Vax=list2(list1('X'),list1('Y'));	e4 = 1;           	Fh = 0;	                   	E  = list2(0,1);	n  = lfirst(F);	U  = list3(E,1,E);	ph = pdegree(1,P);	AL = gfsalgen(p,ph,P);	a1 = F; 	a2 = _0;	while ( a1 != _0 ) {		t  = lfirst(a1);		a1 = lred(a1);		b1 = pmsrem(1,p,lfirst(a1),P);		a1 = lred(a1);		if (b1) a2 = lcomp2(b1,t,a2);	}	a2 = linv(a2);	a2 = pgfsquot(1,p,AL,a2,mp);	a1 = pmsdif(2,p,F,pmsprod(2,p,a2,mp));	a1 = pmsupmsquot(2,p,a1,P);	b1 = _0;	while ( (a1) && (a1 != _0) ) {		t  = lfirst(a1);		a1 = lred(a1);		b2 = pmsrem(1,p,lfirst(a1),P);		a1 = lred(a1);		if (b2) b1 = lcomp2(b2,t,b1);	}	if ( b1 != _0 ) a1 = linv(b1);	else a1 = 0;                 	a1 = upgfsgcd(p,AL,a1,a2);	if ( lfirst(a1) == 0 ) {		U = list1(U);if(neu>4) {printf("\n\n Factor ");putpms(2,p,F,Vax);printf(": polynomial order is local maximal by Dedekind-Criterion.");}		M = oprmsp1basfg(p,F,U);   		M = list1(M);		return(M);	}                          	wy = 1;	y  = 0;	if ( lred2(mp) == _0 ) {   		a1  = list2(0,(p-1));		a0  = list5(E,1,E,0,a1);		ma0 = list4(1,E,0,E);	}	else {		a0  = U;		ma0 = mp;	}	for ( ; ; ) {	    for ( ; ; ) {		if ( e4 == 1 ) {			na = lfirst(ma0);			na = n / na;			if ( na == 1 )  {				y  = 0;				wy = 1;			}			if ( na == wy ) {    				if(y) M = list3(U,a0,y);				else  {				M = list2(U,a0);if(neu>5){printf("\n\n Factor ");putpms(2,p,F,Vax);printf(": Eisenstein-element (1) :");HH = a0; if (HH==0) printf(" 0 # \n"); else {     H2 = lfirst(HH);  printf("\n ( 1 / ");putpms(1,p,H2,list1(list1('X')));printf(")  *   ( ");  H1 = lred(HH);  putpms(2,p,H1,list2(list1('X'),list1('z')));printf(" ) "); }}printf("\n");}				M = oprmsp1basfg(p,F,M);				M = list1(M);if(neu>4){printf("\n\n Factor ");putpms(2,p,F,Vax);printf(": basis for local maximal order:\n");HH = lfirst(M);HH = cdprfmsp1lfm(HH,p);while (HH!=_0) { H1 = lfirst(HH); H1 = uprfmsp1fcdp(p,H1); HH = lred(HH); if (H1==0) printf(" 0 # \n"); else {     H3 = 0;  while ( H1 != _0 ) {	H4=lfirst(H1);	H1=lred(H1);	H2=lfirst(H1);	H1=lred(H1);	if ( !H3 ) printf("\n ");	if ( !oequal(H2,rfmsp1prod(p,H2,H2)) ) { 	 printf("( ");	 putrfmsp1(p,H2,list1(list1('X')));	 printf(" ) ");	 printf(" * ");	}	printf("z^");	putsi(H4);	if ( H1 != _0 ) printf(" + ");	else printf(" ; ");	H3 = 1;  } }}printf("\n");}						return(M);			} 			va = afmsp1regul(p,F,P,Q,a0,ma0,&a1,&a2);			i  = rden(va);			if ( i == na ) {if(neu>5) {H1 = cdprfmsp1sum(p,a0,a2);H2 = iafmsp1mpmpp(p,F,H1,P,1,Q);if ( pdegree(1,H2) == n ) {printf("\n\n Factor ");putpms(2,p,F,Vax);printf(": Eisenstein-element (2) :"); if (H1==0) printf(" 0 # \n"); else {     H2 = lfirst(H1);  printf("\n ( 1 / ");putpms(1,p,H2,list1(list1('X')));printf(")  *   ( ");  H1 = lred(H1);  putpms(2,p,H1,list2(list1('X'),list1('z')));printf(" ) "); }printf("\n");}else {H3 = lfirst(a0);H4 = lred(a0);if ( !pmsrem(1,p,H3,P) ) { H3 = pmsquot(1,p,H3,P); }else H4 = pmsupmsprod(2,p,H4,P);H4 = lcomp(H3,H4);while (1) {H1 = cdprfmsp1sum(p,H1,H4);H2 = iafmsp1mpmpp(p,F,H1,P,1,Q);if ( pdegree(1,H2) == n ) {printf("\n\n Factor ");putpms(2,p,F,Vax);printf(": Eisenstein-element (3) :"); if (H1==0) printf(" 0 # \n"); else {     H2 = lfirst(H1);  printf("\n ( 1 / ");putpms(1,p,H2,list1(list1('X')));printf(")  *   ( ");  H1 = lred(H1);  putpms(2,p,H1,list2(list1('X'),list1('z')));printf(" ) "); }printf("\n");break; }} /* END while */} /* END else  */} /* END if(neu>5) */				M = list3(U,a0,a2);				M = oprmsp1basfg(p,F,M);				M = list1(M);   if(neu>4){printf("\n\n Factor ");putpms(2,p,F,Vax);printf(": basis of local maximal order:\n");HH = lfirst(M);HH = cdprfmsp1lfm(HH,p);while(HH!=_0) { H1 = lfirst(HH); H1 = uprfmsp1fcdp(p,H1); HH = lred(HH); if (H1==0) printf(" 0 # \n"); else {     H3 = 0;  while ( H1 != _0 ) {	H4=lfirst(H1);	H1=lred(H1);	H2=lfirst(H1);	H1=lred(H1);	if ( !H3 ) printf("\n ");	if ( !oequal(H2,rfmsp1prod(p,H2,H2)) ) { 	 printf("( ");	 putrfmsp1(p,H2,list1(list1('X')));	 printf(" ) ");	 printf(" * ");	}	printf("z^");	putsi(H4);	if ( H1 != _0 ) printf(" + ");	else printf(" ; ");	H3 = 1;  } }}printf("\n");}						return(M);			}			Pp = P;			if ( i % wy ) {			    alb = afmsp1idpval(p,F,P,a0,a2,i,y,wy,&y,&wy);if(neu>6){printf("\n v_P^* - value was reduced, y = 1 / %d",wy);printf(" for \n c = "); H1 = y;if (H1==0) printf(" 0 # \n"); else {     H2 = lfirst(H1);  printf(" ( 1 / ");putpms(1,p,H2,list1(list1('X')));printf(")  *   ( ");  H1 = lred(H1);  putpms(2,p,H1,list2(list1('X'),list1('z')));printf(" ) "); }printf("\n was replaced by 1 / kgV(y,z) with z = %d for \n g = ",i); H1 = a0;if (H1==0) printf(" 0 # \n"); else {     H2 = lfirst(H1);  printf(" ( 1 / ");putpms(1,p,H2,list1(list1('X')));printf(")  *   ( ");  H1 = lred(H1);  putpms(2,p,H1,list2(list1('X'),list1('z')));printf(" ) "); }}			}			else {				if ( i == 1 ) Pp = pmsprod(1,p,P,P);				if ( rnum(va) == 1 ) break;     				alb = cdprfmsp1sum(p,a0,a2);							}		}		e4  = 1;                    		ma0 = afmsp1pptf(p,F,P,Q,Pp,a0,alb);		alb = lsecond(ma0);                if (llength(ma0)==2) ma0=lfirst(ma0);		else {			M = lred(ma0);if (neu>2) {HH = lfirst(M);printf("\n\n Core-algorithm: F_i = ");putpms(2,p,F,Vax);printf(" was factored in \n\n ");putpms(2,p,lfirst(HH),Vax);printf("   and \n ");putpms(2,p,lsecond(HH),Vax);printf("\n");}if (neu>3) {HH = lsecond(M);printf("\n Related orthogonal idempotents : \n");while(HH!=_0) { H1 = lfirst(HH); HH = lred(HH); if (H1==0) printf(" 0 # \n"); else {     H2 = lfirst(H1);  printf("\n ( 1 / ");putpms(1,p,H2,list1(list1('X')));printf(")  *   ( ");  H1 = lred(H1);  putpms(2,p,H1,Vax);printf(" ) "); }printf("\n");}}					return(M);		}		a0  = alb;	    }	    y   = a2;	    wy  = rden(va);	    clb = list3(E,0,E);	    B   = list2(U,a0);	    B   = oprmsp1basfg(p,F,B);	    B   = cdprfmsp1lfm(B,p);	    Ph  = pmsprod(1,p,P,P);	    b0  = afmsp1expsp(p,F,a2,wy,Ph);	    b0  = cdprfmsp1upq(p,b0,P);	    if ( modprmsp1elt(B,p,b0) == 0 ) { 		a1  = a2;		Pp  = P;		MOD = E;	    }	    else {                		B   = list1(a0);		B   = oprmsp1basfg(p,F,B);		B   = cdprfmsp1lfm(B,p);		B2  = B;		MOD = E;		while ( B2 != _0 ) {			Bl  = lfirst(B2);			B2  = lred(B2);			MOD = pmsprod(1,p,MOD,lthird(Bl));		}                         		Ph = pmsquot(1,p,Pp,upmsgcd(p,Pp,MOD));		Ph = pmsprod(1,p,MOD,Ph);		a1 = cdprfmsp1mh(p,a1,Ph);                Pp = P;		Ph = pmsquot(1,p,P,upmsgcd(p,P,MOD));		Ph = pmsprod(1,p,MOD,Ph);		Ph = pmsprod(1,p,Ph,P);		b0 = afmsp1expsp(p,F,a1,wy,Ph);		b0 = cdprfmsp1upq(p,b0,P);	    }                                           	    if ( !Fh ) {	        B2 = F;	        Bl = _0;	        while ( B2 != _0 ) {			t  = lfirst(B2);			B2 = lred(B2);			R2 = list2(lfirst(B2),E);			B2 = lred(B2);			Bl = lcomp2(R2,t,Bl);		}	        Fh = linv(Bl);		    }	    for ( ; ; ) {                         		mb0 = afmsp1pptf(p,F,P,Q,P,a0,b0);		b1  = lsecond(mb0);                if (llength(mb0)==2) mb0 = lfirst(mb0);		else {                     			M = lred(mb0);if (neu>2) {HH = lfirst(M);printf("\n\n Core-algorithm : F_i = ");putpms(2,p,F,Vax);printf(" factored in \n\n ");putpms(2,p,lfirst(HH),Vax);printf("   and \n ");putpms(2,p,lsecond(HH),Vax);printf("\n");}if (neu>3) {HH = lsecond(M);printf("\n Related orthogonal idempotents : \n");while(HH!=_0) { H1 = lfirst(HH); HH = lred(HH); if (H1==0) printf(" 0 # \n"); else {     H2 = lfirst(H1);  printf("\n ( 1 / ");putpms(1,p,H2,list1(list1('X')));printf(")  *   ( ");  H1 = lred(H1);  putpms(2,p,H1,Vax);printf(" ) "); }printf("\n");}}			return(M);		}		Ph  = pmsquot(1,p,P,upmsgcd(p,P,MOD));		Ph  = pmsprod(1,p,MOD,Ph);		b0  = cdprfmsp1mh(p,b0,Ph);		alb = cdprfmsp1sum(p,a0,b0);		i   = lfirst(ma0);		r   = lfirst(mb0);		if ( i % r ) {			e4 = 0;if(neu>6){printf("\n\n Core-algorithm: a and b with f_a do not divide f_b.");printf("\n Replace a by a+b.");printf("\n a = "); H1 = a0;if (H1==0) printf(" 0 # \n"); else {     H2 = lfirst(H1);  printf(" ( 1 / ");putpms(1,p,H2,list1(list1('X')));printf(")  *   ( ");  H1 = lred(H1);  putpms(2,p,H1,list2(list1('X'),list1('z')));printf(" ) "); }printf("\n b = "); H1 = b0;if (H1==0) printf(" 0 # \n"); else {     H2 = lfirst(H1);  printf(" ( 1 / ");putpms(1,p,H2,list1(list1('X')));printf(")  *   ( ");  H1 = lred(H1);  putpms(2,p,H1,list2(list1('X'),list1('z')));printf(" ) "); }printf("\n f_a = %d, f_b = %d.",i,r);}			break;		}	 	vb = afmsp1regul(p,F,P,Q,b0,mb0,&b1,&b2);		i  = rden(va);		r  = rden(vb);		if ( i % r ) {                             			alb = afmsp1idpval(p,F,P,a0,a2,i,b2,r,&y,&wy);if(neu>6){printf("\n v_P^* - value was reduced, y = 1 / %d",r);printf(" for \n c = "); H1 = b0;if (H1==0) printf(" 0 # \n"); else {     H2 = lfirst(H1);  printf(" ( 1 / ");putpms(1,p,H2,list1(list1('X')));printf(")  *   ( ");  H1 = lred(H1);  putpms(2,p,H1,list2(list1('X'),list1('z')));printf(" ) "); }printf("\n was replaced by 1 / kgV(y,z) with z = %d for \n g = ",i); H1 = a0;if (H1==0) printf(" 0 # \n"); else {     H2 = lfirst(H1);  printf(" ( 1 / ");putpms(1,p,H2,list1(list1('X')));printf(")  *   ( ");  H1 = lred(H1);  putpms(2,p,H1,list2(list1('X'),list1('z')));printf(" ) "); }}			e4 = 0;			break;		}              		B2 = afmsp1pptf(p,F,P,Q,P,a0,alb);		Bl = lsecond(B2);                if (llength(B2)==2) B2=lfirst(B2);		else {			M = lred(B2);if (neu>2) {HH = lfirst(M);printf("\n\n Core-algorithm : F_i = ");putpms(2,p,F,Vax);printf(" factored in \n\n ");putpms(2,p,lfirst(HH),Vax);printf("   and \n ");putpms(2,p,lsecond(HH),Vax);printf("\n");}if (neu>3) {HH = lsecond(M);printf("\n Related orthogonal idempotents : \n");while(HH!=_0) { H1 = lfirst(HH); HH = lred(HH); if (H1==0) printf(" 0 # \n"); else {     H2 = lfirst(H1);  printf("\n ( 1 / ");putpms(1,p,H2,list1(list1('X')));printf(")  *   ( ");  H1 = lred(H1);  putpms(2,p,H1,Vax);printf(" ) "); }printf("\n");}}					return(M);				}         		R2  = pmsprod(1,p,Q,MOD);   		e   = upmsaddval(p,P,R2);                                      e   = iprod(e,n);		ph2 = iprod(ph,lfirst(ma0));		ph2 = iexp(p,ph2);		ph3 = ph2;		while ( icomp(e,ph3) == 1 ) ph3 = iprod(ph3,ph2);		blb = afmsp1expsp(p,F,b0,ph3,R2);		Ph  = pmsquot(1,p,P,upmsgcd(p,P,MOD));		Ph  = pmsprod(1,p,MOD,Ph);		blb = cdprfmsp1mh(p,blb,Ph);		B2  = pmsneg(2,p,lred(b0));		lsred(b0,B2);		B2  = cdprfmsp1sum(p,blb,b0);		c0  = afmsp1prodsp(p,F,B2,clb,0);		mc0 = list2(1,E);		vc  = afmsp1regul(p,F,P,Q,c0,mc0,&c1,&c2);		t   = rfloor(vc);		t   = t + 1;		P2  = pmsexp(1,p,P,t);		Ph  = pmsquot(1,p,P2,upmsgcd(p,P2,MOD));		R2  = pmsprod(1,p,Ph,MOD);		c0  = cdprfmsp1mh(p,c0,R2); 		i   = rden(va);		r   = rden(vc);		if ( i % r ) {			alb = afmsp1idpval(p,F,P,a0,a2,i,c2,r,&y,&wy);if(neu>6){printf("\n v_P^* - value was reduced, y = 1 / %d",r);printf(" for\n c = "); H1 = c0;if (H1==0) printf(" 0 # \n"); else {     H2 = lfirst(H1);  printf(" ( 1 / ");putpms(1,p,H2,list1(list1('X')));printf(")  *   ( ");  H1 = lred(H1);  putpms(2,p,H1,list2(list1('X'),list1('z')));printf(" ) "); }printf("\n was replaced by 1 / kgV(y,z) with z = %d for \n g = ",i); H1 = a0;if (H1==0) printf(" 0 # \n"); else {     H2 = lfirst(H1);  printf(" ( 1 / ");putpms(1,p,H2,list1(list1('X')));printf(")  *   ( ");  H1 = lred(H1);  putpms(2,p,H1,list2(list1('X'),list1('z')));printf(" ) "); }}			e4 = 0;			break;		}		B2  = rquot(vc,va);		k   = rnum(B2);		clb = afmsp1expsp(p,F,a1,k,R2);                Ph  = uprfmsp1fcdp(p,clb);		b0  = list2(P2,E);		b0  = list2(0,b0);		do {			B2 = uprfmsp1egcd(p,Fh,Ph,&Bl,&R2);			if ( lfirst(B2) ) Ph = prfmsp1sum(1,p,Ph,b0);		} while ( lfirst(B2) );                         		clb = cdprfmsp1fup(p,Ph);         		Bl  = cdprfmsp1inv(p,Fh,clb);		ph2 = upmsaddval(p,P,MOD);		R2  = pmsexp(1,p,P,ph2);		R2  = pmsquot(1,p,MOD,R2);		Ph  = lfirst(Bl);		Bl  = lred(Bl);		ph2 = upmsaddval(p,P,Ph);		Ph  = pmsexp(1,p,P,ph2);		Bl  = pmsupmsprod(2,p,Bl,R2);		Bl  = lcomp(Ph,Bl);		b0  = afmsp1prodsp(p,F,c0,Bl,0);	    }	}}