(kill(all), algebraic : true, 0);
0$

ratsimp([2 / sqrt(2), -2/sqrt(2), 1 / (1 + sqrt(2)), -1/(1 + sqrt(2)), 1/(sqrt(2) - 1), -1/(sqrt(2) -1)]);
[sqrt(2),  -sqrt(2), sqrt(2) - 1, 1 - sqrt(2), sqrt(2) + 1,  -1 - sqrt(2)]$

ratsimp([1/(sqrt(5) - sqrt(2)), 1/(sqrt(5) + sqrt(2))]);
[((sqrt(5) + sqrt(2))/3), ((sqrt(5) - sqrt(2))/3)]$

ratsimp(1/(5^(1/3) - 2^(1/3)));
((5^(2/3) + 2^(1/3) * 5^(1/3) + 2^(2/3))/3)$

ratsimp(-1/(5^(2/3) + 2^(2/3)));
((2^(2/3) * 5^(2/3) - 5 * 5^(1/3) - 2 * 2^(1/3))/29)$

(eq : 4*z^2-2*z-1,0);
0$

(sol : solve(eq,z),0);
0$

map(lambda([s], ratsimp(subst(s,eq))), sol);
[0,0]$

(eq : 4*z^4-2*z^2-1,0);
0$

(sol : solve(eq,z),0);
0$

map(lambda([s], ratsimp(subst(s,eq))), sol);
[0,0,0,0]$

(eq : 4*z^3-2*z^2-1,0);
0$

(sol : solve(eq,z),0);
0$

map(lambda([s], ratsimp(subst(s,eq))), sol);
[0,0,0]$

(eq : 4*z^3-2*z^2-sqrt(42),0);
0$

(sol : solve(eq,z),0);
0$

map(lambda([s], ratsimp(subst(s,eq))), sol);
[0,0,0]$

(eq : 4*z^3 + (12 + %pi * %i) *z^2-sqrt(42),0);
0$

(sol : solve(eq,z),0);
0$

map(lambda([s], ratsimp(subst(s,eq))), sol);
[0,0,0]$

(eq : 4*z^4 + z - 1,0);
0$

(sol : solve(eq,z),0);
0$

/* This needs some extra help -- I don't consider this to be a bug. */

map(lambda([s], ratsimp(radcan(subst(s,eq)))), sol);
[0,0,0,0]$

/* By the way, cos(%pi/32) is a root of this polynomial.*/

(eq : 32768*x^16-131072*x^14+212992*x^12-180224*x^10+84480*x^8-21504*x^6+2688*x^4-128*x^2+1,0);
0$

(sol : solve(eq,x),0);
0$

map(lambda([s], rectform(ratsimp(subst(s,eq)))), sol);
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]$

/* By the way, cos(%pi/60) is a root of this polynomial. 
Also, cos(%pi/60) = sqrt(sqrt(15)+sqrt(2)*sqrt(5-sqrt(5))+sqrt(3)+8)/4 */

(eq : 65536*z^16-262144*z^14+430080*z^12-372736*z^10+182784*z^8-50176*z^6+7040*z^4-384*z^2+1,0);
0$

(sol : solve(eq,z),0);
0$

map(lambda([s], rectform(ratsimp(subst(s,eq)))), sol);
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]$
 
ratsimp((1 + %i)/(1 - %i));
%i$

ratsimp(1/(7 + %i * 7));
 - ((%i - 1)/14)$

(tellrat(x,x^2+5*x+1),0);
0$

ratsimp([0,1,x,x^2,x^3]);
[0,1,x,-5*x-1,24*x+5]$

ratsimp([0,1,1/x,1/x^2,1/x^3]);
[0, 1,  - x - 5, 5 * x + 24,  - 24 * x - 115]$

ratsimp(a + b * x + c * x^2 + d * x^3);
(24*d-5*c+b)*x+5*d-c+a$

ratsimp((a + b * x + c * x^2 + d * x^3) * ratsimp(1/(a + b * x + c * x^2 + d * x^3)));
1$

ratsimp([%i, minf, inf, infinity]);
[%i, minf, inf, infinity]$

ratsimp(makelist(%i^k,k,-8,8));
[1,%i,-1,-%i,1,%i,-1,-%i,1,%i,-1,-%i,1,%i,-1,-%i,1]$

ratsimp([1/(x+1), 1/(x -1), 1/(x - 1)^2]);
[((x + 4)/3),  -((x + 6)/7), ((x + 5)/7)]$

ratsimp(cos(x^2));
cos(5*x+1)$

ratsimp(matrix([1,x],[x^2,x^3]));
matrix([1,x],[-5*x-1, 24 * x + 5])$

(untellrat(x),0);
0$

ratsimp([1,x,x^2,x^3]);
[1,x,x^2,x^3]$

ratsimp([1,1/x,1/x^2,1/x^3]);
[1,1/x,1/x^2,1/x^3]$

(reset(algebraic), remvalue(eq, sol),0);
0$