>
> iload "ex3-1.m";
Interactive-loading "ex3-1.m"
> // Define the quaternion algebra:
> // p equiv 3 mod 4
> p := 67;
> B := QuaternionAlgebra< RationalField() | -1, -p >;
> _, i, j, k := Explode(Basis(B));
> i^2 eq -1;
true
> j^2 eq -p;
true
> RamPrimes := RamifiedPrimes(B); RamPrimes;
[ 67 ]
> D := Discriminant(B); D;
67
> for q in RamPrimes do
for>   assert D mod q eq 0;
for> end for;
> // p not 3 mod 4:
> // need to find a small primes r with r = 3 mod 4
> // and r a non-square mod p
> p := 41;
> r := 3;
> while (r mod 4 ne 3) or KroneckerSymbol(-r, p) ne -1 do
while>   r := NextPrime(r);
while> end while;
> print r;
3
> B<i,j, k> := QuaternionAlgebra< RationalField() | -r, -p >;
> Basis(B);
[ 1, i, j, k ]
> i^2 eq -r;
true
> j^2 eq -p;
true
> // Same check of ramification
> RamPrimes := RamifiedPrimes(B); RamPrimes;
[ 41 ]
> D := Discriminant(B); D;
41
> for q in RamPrimes do
for>   assert D mod q eq 0;
for> end for;
> /// define the element
> //  2 + i ^"^╚^╥ 3j + 4k
> w := B![2, 1, -3, 4];
> // or w := 2 + i - 3*j +4k;
> Norm(w), Trace(w), MinimalPolynomial(w);
2344 4 $.1^2 - 4*$.1 + 2344
>