gh-37225: splitting_field: polred -> polredbest
    
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This PR partially addresses
#22165. It removes one (in
`splitting_field`) of the two uses of the Pari deprecated function
`polred`. The other one is less trivial to remove and should probably be
discussed in the issue.

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### ⌛ Dependencies
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URL: #37225
Reported by: AurelPage
Reviewer(s): Vincent Delecroix

Author: Release Manager

src/sage/rings/number_field/splitting_field.py

@@ -147,7 +147,7 @@ def splitting_field(poly, name, map=False, degree_multiple=None, abort_degree=No
 
     - ``simplify`` -- (default: ``True``) during the algorithm, try
       to find a simpler defining polynomial for the intermediate
-      number fields using PARI's ``polred()``.  This usually speeds
+      number fields using PARI's ``polredbest()``.  This usually speeds
       up the computation but can also considerably slow it down.
       Try and see what works best in the given situation.
 
@@ -549,11 +549,9 @@ def splitting_field(poly, name, map=False, degree_multiple=None, abort_degree=No
             # Find a simpler defining polynomial Lpol for Mpol
             verbose("New field before simplifying: %s" % Mpol, t)
             t = cputime()
-            M = Mpol.polred(flag=3)
-            n = len(M[0])-1
-            Lpol = M[1][n].change_variable_name("y")
-            LtoM = M[0][n].change_variable_name("y").Mod(Mpol.change_variable_name("y"))
-            MtoL = LtoM.modreverse()
+            M = Mpol.polredbest(1)
+            Lpol = M[0].change_variable_name("y")
+            MtoL = M[1].lift().change_variable_name("y").Mod(Lpol)
         else:
             # Lpol = Mpol
             Lpol = Mpol.change_variable_name("y")