diff --git a/src/sage/rings/valuation/augmented_valuation.py b/src/sage/rings/valuation/augmented_valuation.py
index 5b4666d45cb..c02b0a6957c 100644
--- a/src/sage/rings/valuation/augmented_valuation.py
+++ b/src/sage/rings/valuation/augmented_valuation.py
@@ -433,6 +433,12 @@ def augmentation_chain(self):
         r"""
         Return a list with the chain of augmentations down to the underlying :mod:`Gauss valuation <sage.rings.valuation.gauss_valuation>`.
 
+        .. NOTE::
+
+            This method runs in time linear in the length of the chain (though
+            the printed representation might seem to indicate that it takes
+            quadratic time to construct the chain.)
+
         EXAMPLES::
 
             sage: R.<x> = QQ[]
diff --git a/src/sage/rings/valuation/developing_valuation.py b/src/sage/rings/valuation/developing_valuation.py
index a73130c9cea..7b5935fca35 100644
--- a/src/sage/rings/valuation/developing_valuation.py
+++ b/src/sage/rings/valuation/developing_valuation.py
@@ -40,7 +40,7 @@
     [x + 1, 1]
 """
 # ****************************************************************************
-#       Copyright (C) 2013-2017 Julian Rüth <julian.rueth@fsfe.org>
+#       Copyright (C) 2013-2025 Julian Rüth <julian.rueth@fsfe.org>
 #
 #  Distributed under the terms of the GNU General Public License (GPL)
 #  as published by the Free Software Foundation; either version 2 of
@@ -102,6 +102,29 @@ def phi(self):
             sage: v = GaussValuation(S)                                                 # needs sage.libs.ntl
             sage: v.phi()                                                               # needs sage.libs.ntl
             (1 + O(2^5))*x
+
+        Use
+        :meth:`~sage.rings.valuation.inductive_valuation.InductiveValuation.augmentation_chain`
+        to obtain the sequence of key polynomials of an
+        :class:`~sage.rings.valuation.inductive_valuation.InductiveValuation`::
+
+            sage: R.<x> = QQ[]
+            sage: v = GaussValuation(R, QQ.valuation(2))
+            sage: v = v.augmentation(x, 1)
+            sage: v = v.augmentation(x^2 + 2*x + 4, 3)
+
+            sage: v
+            [ Gauss valuation induced by 2-adic valuation, v(x) = 1, v(x^2 + 2*x + 4) = 3 ]
+
+            sage: [w.phi() for w in v.augmentation_chain()[:-1]]
+            [x^2 + 2*x + 4, x]
+
+        A similar approach can be used to obtain the key polynomials and their
+        corresponding valuations::
+
+            sage: [(w.phi(), w.mu()) for w in v.augmentation_chain()[:-1]]
+            [(x^2 + 2*x + 4, 3), (x, 1)]
+
         """
         return self._phi