Fix bug in decomposition_type

src/sage/rings/number_field/number_field.py

@@ -5998,12 +5998,21 @@ def decomposition_type(self, p):
             2
             sage: M.decomposition_type(Q1)
             [(2, 5, 1)]
+
+        Check that :trac:`34514` is fixed::
+
+            sage: K.<a> = NumberField(x^4 + 18*x^2 - 1)
+            sage: R.<y> = K[]
+            sage: L.<b> = K.extension(y^2 + 9*a^3 - 2*a^2 + 162*a - 38)
+            sage: [L.decomposition_type(i) for i in K.primes_above(3)]
+            [[(1, 1, 2)], [(1, 1, 2)], [(1, 2, 1)]]
         """
         v0 = self.base_ring().valuation(p)
         e0 = v0.value_group().gen().denominator()
-        f0 = v0.residue_field().degree()
+        # Ideally we would compute f using the degree, but residue fields of relative extensions are currently implemented using polynomial quotient rings (this will hopefully be improved after #28485).
+        C0 = v0.residue_field().cardinality()
         valuations = v0.extensions(self)
-        ef = [(v.value_group().gen().denominator() // e0, v.residue_field().degree() // f0) for v in valuations]
+        ef = [(v.value_group().gen().denominator() // e0, v.residue_field().cardinality().exact_log(C0)) for v in valuations]
         return sorted([(e, f, g) for ((e, f), g) in Counter(ef).items()])
 
     def gen(self, n=0):