gh-37111: Add test if quaternion order is maximal

    
Add a method to sage.algebras.quatalg.quaternion_algebra.QuaternionOrder
which allows to check whether the order is maximal.
This method compares the discriminant of the order to the discriminant
of the algebra. According to Voight's book on quaternion algebras
(chapter 15), this test is valid in number fields (but not over any
field), so the method fails with a notImplementedError in  other cases.

Currently, there is no method to test maximality of orders, so
implementing it for quaternion algebras over number fields is some
progress.


#sd123
    
URL: #37111
Reported by: syndrakon
Reviewer(s): David Coudert, Peter Bruin, syndrakon

src/sage/algebras/quatalg/quaternion_algebra.py

@@ -77,6 +77,7 @@
 from sage.misc.cachefunc import cached_method
 
 from sage.categories.algebras import Algebras
+from sage.categories.number_fields import NumberFields
 
 ########################################################
 # Constructor
@@ -1783,6 +1784,42 @@ def discriminant(self):
 
         return (MatrixSpace(QQ, 4, 4)(L)).determinant().sqrt()
 
+    def is_maximal(self):
+        r"""
+        Check whether the order of ``self`` is maximal in the ambient quaternion algebra.
+
+        Only works in quaternion algebras over number fields
+
+        OUTPUT: Boolean
+
+        EXAMPLES::
+
+            sage: p = 11
+            sage: B = QuaternionAlgebra(QQ, -1, -p)
+            sage: i, j, k = B.gens()
+            sage: O0_basis = (1, i, (i+j)/2, (1+i*j)/2)
+            sage: O0 = B.quaternion_order(O0_basis)
+            sage: O0.is_maximal()
+            True
+            sage: O1 = B.quaternion_order([1, i, j, i*j])
+            sage: O1.is_maximal()
+            False
+
+        TESTS::
+
+            sage: B = QuaternionAlgebra(GF(13), -1, -11)
+            sage: i, j, k = B.gens()
+            sage: O0_basis = (1, i, j, k)
+            sage: O0 = B.quaternion_order(O0_basis)
+            sage: O0.is_maximal()
+            Traceback (most recent call last):
+            ...
+            NotImplementedError: check for maximality is only implemented for quaternion algebras over number fields
+        """
+        if self.quaternion_algebra().base_ring() not in NumberFields():
+            raise NotImplementedError("check for maximality is only implemented for quaternion algebras over number fields")
+        return self.discriminant() == self.quaternion_algebra().discriminant()
+
     def left_ideal(self, gens, check=True, *, is_basis=False):
         r"""
         Return the left ideal of this order generated by the given generators.