\\ Q(x)/x^2
mt1 = [matid(2), [0,0; 1,0]];
alisassociative(mt1)
al1 = altableinit(mt1);

\\Matrices [*,*;0,*] in F2
mt2 = [[1,0,0;0,1,0;0,0,1],[0,0,0;1,0,1;0,0,0],[0,0,0;0,0,0;1,0,1]];
alisassociative(mt2,2)
al2 = altableinit(mt2,2);

aliscommutative(al1)
aliscommutative(al2)
alcenter(al2)

algetdim(al1)
algetdim(al2)
algetchar(al1)
algetchar(al2)
algetmultable(al1)
algetmultable(al2)

x1 = [1,2]~;
y1 = [-1,3]~;
x2 = [1,1,0]~;
y2 = [0,0,1]~;
alneg(al1, x1)
aladd(al1, x1, y1)
alsub(al2, x2, y2)
almul(al2, x2, y2)
alsqr(al1, x1)
alpow(al1, x1, 7)
alinv(al2, x2)
alisinv(al2, y2)
alisinv(al1, x1, &ix1)
ix1
almultable(al2, y2)
altrace(al2, x2)
alnorm(al2, x2)
alcharpoly(al2, x2)

alissemisimple(al1)
alissemisimple(al2)
rad1 = alradical(al1)
rad2 = alradical(al2)
al3 = alquotient(al1, rad1);
alissemisimple(al3)
al4 = alquotient(al2, rad2);
alissemisimple(al4)

alissimple(al3)
alissimple(al4)
dec = alsimpledec(al4);
#dec

\\quaternion algebra
pol = y^2-5;
nf = nfinit(pol);
al5 = alinit(nf, [-1,y]);
pol2 = algetsplitting(al5).pol;
x5 = [x+y, 2*x]~*Mod(1,pol)*Mod(1,pol2);
y5 = alalgtobasis(al5, x5)
albasistoalg(al5, y5)
altrace(al5, x5)
alnorm(al5, x5)
alcharpoly(al5, x5, t)

\\cyclic algebra
Q = nfinit(y);
pol = polcyclo(5,'x)
rnf = rnfinit(Q, pol);
al6 = alinit(rnf, [Mod(x^2,pol), 3]);

algetcenter(al6)
algetdegree(al5)
algetdegree(al6)
algetdim(al6)
algetabsdim(al6)

algethassef(al5)
algethassei(al5)

algethassef(al6)
algethassei(al6)

\\algebra given by Hasse invariants
pr3 = idealprimedec(nf, 3)[1];
pr11 = idealprimedec(nf, 11)[1];
hi = [0,0];
hf = [[pr3,pr11], [1/3, 2/3]];
al7 = alinit(nf, [3, hf, hi]);
algethassef(al7)
algethassei(al7)