P = 1 + x^2 + 27*x^10
Q = sum(i = 1, 10, (i+1) * x^i)
v=[1,2,3,4];
T=Pol(v)
U=Polrev(v)
polrecip(T)==U

f=3*x^2+x*y+4*y;
polcoef(f,1)
d=poldegree(f);d
pollead(f)==polcoef(f,d)
polcoef(f,1,y)
dy=poldegree(f,y);dy
pollead(f,y)==polcoef(f,d,y)

P=x^3 - 6*x^2 + 11*x - 6;
polroots(P)
algdep(3,2)
z = 2^(1/6)+3^(1/5);
algdep(z, 30)
a=[1,2,3];b=[4,0,2];
polinterpolate(a,b)
polfromroots(a)

f= x^5 + x^4 + 5*x^3 + 3*x^2 + 3*x - 1;
factor(f)
pol = x^4 - 4*x^2 + 16
polisirreducible(pol)
factor(pol)
factor(poldisc(pol))
factorpadic(pol, 3, 10)
factorpadic(pol, 11, 10)
factormod(pol,2)
factormod(pol,11)
lift(%)

g = polcyclo(30)
Mod(x,f)^5
lift(Mod(x,g)^15)

pol2=x^4 - 4*x^2 + 16
fn = lift( factornf(pol2, t^2 + 1) )

T=(x^2-1)^3-2; polgalois(T)
P= x^8-7*x^6+14*x^4-8*x^2+1
polgalois(P)

gal=galoisinit(P);
gal2=galoissplittinginit(T);
poldegree(gal2.pol)

gal.pol
gal.roots
gal.group
gal.gen

L = galoissubgroups(gal);
  vector(#L, i, galoisisabelian(L[i],1))
  vector(#L, i, galoisidentify(L[i]))
[a,b]=%[1]; galoisgetname(a,b)
galoisfixedfield(gal,L[1])
galoisfixedfield(gal,L[2])
galoissubfields(gal);
 vector(#%, i, %[i][1])

p = randomprime(2^100)
a = Mod(2,p);
type(a)
a^(p-1)
a.mod == p
lift(a)

T = x^2+1;
b = Mod(x+a, T);
type(b)
b.pol
b.mod == T

c = ffgen(3^8,'c)
type(c)
c.p
c.mod
polisirreducible(c.mod*Mod(1,3))
c.f

d = c^9+1
d.pol
type(d.pol)
ffinit(3,5)
ffgen(x^2+Mod(1,3))

[c,c+1;2*c,1]^-1
d = random(c)
issquare(d)
trace(d)
norm(d)
minpoly(d^82)

factor(x^5+x^3+c)
polrootsmod(x^7+x+c)