Finding the discriminant of a quaternion algebra

algebraic-number-theoryalgebrasdiscriminantquaternions

Consider the totally real number field $ F=\mathbb{Q}(\zeta_{10}+\zeta_{10}^*) $. Consider the quaternion algebra $ Q=(\frac{-1,-1}{F}) $. How do I compute the discriminant of this algebra?

I gave this example, but any example of computing a discriminant of a quaternion algebra like this, where $ F $ is a totally real number field of degree at least $ 3 $, would be useful.

Background:

The textbook by John Voight on Quaternion Algebras says that the definition of the discriminant of an order in a Quaternion algebra is
$$
det(trd(\alpha_i \alpha_j)_{i,j=1,2,3,4})
$$

Finding the discriminant of the rational Hamiltonian Quaternions $ B=(-1,-1|\mathbb{Q}) $:

I'm not sure how to prove this, but the Hurwitz quaternions $ \mathcal{O} $ are a maximal order in $ B $.

A basis for $ \mathcal{O} $ is $ 1,i,j,\frac{1+i+j+ij}{2} $. So the upper $ 3 \times 3 $ block of the matrix $ trd(\alpha_i \alpha_j)_{i,j=1,2,3,4} $ is same as in example 15.2.10 since the first three elements of the basis are the same as for the standard (Lipschitz) order. Thus the top $ 3 \times 3 $ block is
\begin{pmatrix}
2 & 0 & 0 & \\
0 & 2a & 0 & \\
0 & 0 & 2b & \\
& & &
\end{pmatrix}
and the full matrix is
\begin{pmatrix}
2 & 0 & 0 & 1 \\
0 & 2a & 0 & a \\
0 & 0 & 2b & b\\
1 & a & b & \frac{1+a+b-ab}{2}
\end{pmatrix}
which has determinant $ -4a^2b^2 $. Note that this is different from the discriminant of the standard order which is $ -16a^2b^2 $.

Returning to the original problem, my idea so far is:

Let $ R_F=\mathbb{Z}[\zeta_{10}+\zeta_{10}^*] $. Something analogous to Lipschitz quaternions like
$$
a+bi+cj+d ij, \; a,b,c,d \in R_F
$$
is certainly an order, but to get a maximal order perhaps I want an order analogous to the Hurwitz quaternions like
$$
a+bi+cj+d ij, \; a,b,c,d \in R_F \text{ or } a,b,c,d \in R_F+\frac{1}{2}
$$

But how do I know if that order is even maximal?

Best Answer

Here's code that you can run in the Magma calculator (http://magma.maths.usyd.edu.au/calc/):

K<z> := CyclotomicField(10);
F := sub<K | z+1/z>;
B<i,j> := QuaternionAlgebra(F, -1, -1);
omega := (-1+i+j+i*j)/2;
O := Order([1, i, j, omega]);
IsMaximal(O);
Discriminant(B);
Discriminant(MaximalOrder(B));
Discriminant(O);

It returns:

false
Principal Ideal
Generator:
    [1, 0]
[ 1st place at infinity, 2nd place at infinity ]
Principal Ideal
Generator:
    [1, 0]
Principal Prime Ideal
Generator:
    [2, 0]

This says that the quaternion algebra has discriminant $(1)$, of course equal to the (reduced) discriminant of a maximal order. However, as you computed, the Hurwitz order over $\mathbb{Z}$ has (reduced) discriminant $2$, so by the same matrix calculation it has the same discriminant over $R_F$. Therefore, it is not a maximal order.

There is a lot implemented (http://magma.maths.usyd.edu.au/magma/handbook/quaternion_algebras) and I encourage you to play around! For example, you can find out a (pseudo)basis for a maximal order, compute the Eichler invariant of the order, compute an embedding into a matrix ring, etc.!

Final comment: as Kimball was suggesting, this is a local question so knowing the reduced discriminant of an order, you can check if it is maximal by checking that it exactly the product of the primes that ramify using the Hilbert symbol.

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