More detailed NOTE for any_root()

Author: GiacomoPope

src/sage/rings/polynomial/polynomial_element.pyx

@@ -2309,15 +2309,6 @@ cdef class Polynomial(CommutativePolynomial):
           used for polynomials over finite fields. If not ``None`` only returns
           irreducible factors of ``self`` whose degree divides ``ext_degree``.
 
-        .. NOTE::
-
-            Any root is non-deterministic when finding linear roots of a
-            polynomial over the base ring. However, if ``degree`` is greater
-            than one, or ``ring`` is an extension of the base ring, then
-            eventually the root is found by returning a single root after
-            factorisation. Roots found in this way are deterministic.
-            This may change in the future.
-
         EXAMPLES::
 
             sage: # needs sage.rings.finite_rings
@@ -2482,6 +2473,17 @@ cdef class Polynomial(CommutativePolynomial):
             Negative degree input will be deprecated. Instead use
             ``assume_equal_deg``.
 
+        .. NOTE::
+
+            For finite fields, ``any_root()`` is non-deterministic when
+            finding linear roots of a polynomial over the base ring.
+            However, if ``degree`` is greater than one, or ``ring`` is an
+            extension of the base ring, then eventually the root is found
+            by returning a single root after factorisation. Roots found
+            in this way are deterministic. This may change in the future.
+            For all other rings or fields, roots are found by first
+            fully-factoring the polynomial and the output is deterministic.
+
         EXAMPLES::
 
             sage: # needs sage.rings.finite_rings