# Square-free factorization.

f := expand((x^3+x+4)^2*(x+1));

f_prime := diff(f,x);

print(`gcd(f,f_prime) is `, gcd(f,f_prime));
g := gcd(f,f_prime);
print(`f1 is quo(f,g^2,x,'r') = `, quo(f,g^2,x,'r'));
f1 := quo(f,g^2,x,'r');
print(`r is `, r);

print(` g * g * f1 is `, g*g*f1);
sort(expand(g*g*f1));
sort(f);


print(`gcd(diff(g,x),g) is `, 
	gcd(diff(g,x),g));

#  This is 1, so g is square-free.

print(`------`);

f := expand((2*x+1)^14) mod 7;
print(`diff(f,x) is `,
	diff(f,x));
print(` diff(f,x) mod 7 is`, diff(f,x) mod 7);
print(`gcd(f,diff(f,x)mod 7) is `, gcd(f,diff(f,x)mod 7) );

f_7th_root := 4*x^2 + 4*x + 1;

print(` expand(f_7th_root ^7) mod 7   is  `,
	expand(f_7th_root ^7) mod 7);

f_7 :=  f_7th_root;
diff(f_7,x);
" mod 7;
print(` gcd(f_7,diff(f_7,x) mod 7,x) is`,
	gcd(f_7,diff(f_7,x) mod 7,x));
#print(`quo(f_7,g,x,'r') is `,
#	quo(f_7,g,x,'r') );
#print(` remainder r is `, r);