Trac #28976: AssertionError when creating valuations from prime ideals

{{{
R.<x> = QQ[]
K.<theta_K> = NumberField(x^6 - 18*x^4 - 24*x^3 + 27*x^2 + 36*x - 6)
fp = K.fractional_ideal((2, -7/44*theta_K^5 + 19/44*theta_K^4 +
87/44*theta_K^3 - 87/44*theta_K^2 - 5/2*theta_K + 39/22))
print(fp.norm())               # yields 2
print(fp in K.primes_above(2)) # yields True
v = K.valuation(fp)            # raises AssertionError
}}}

see https://groups.google.com/forum/#!topic/sage-nt/zEkLa-
4cgys/discussion

{{{
K.<pi, w> = NumberField([x^2 - 2, x^2 + x + 1]);
K.valuation(pi) # raises AssertionError
}}}

see https://groups.google.com/forum/#!topic/sage-nt/zEkLa-
4cgys/discussion

URL: https://trac.sagemath.org/28976
Reported by: saraedum
Ticket author(s): Julian Rüth
Reviewer(s): Vincent Delecroix

Author: Release Manager

src/sage/rings/padics/padic_valuation.py

@@ -35,7 +35,7 @@
 
 """
 #*****************************************************************************
-#       Copyright (C) 2013-2018 Julian Rüth <[email protected]>
+#       Copyright (C) 2013-2020 Julian Rüth <[email protected]>
 #
 #  Distributed under the terms of the GNU General Public License (GPL)
 #  as published by the Free Software Foundation; either version 2 of
@@ -285,6 +285,26 @@ def create_key_and_extra_args_for_number_field_from_ideal(self, R, I, prime):
             sage: GaussianIntegers().valuation(GaussianIntegers().ideal(2)) # indirect doctest
             2-adic valuation
 
+        TESTS:
+
+        Verify that :trac:`28976` has been resolved::
+
+            sage: R.<x> = QQ[]
+            sage: K.<a> = NumberField(x^6 - 18*x^4 - 24*x^3 + 27*x^2 + 36*x - 6)
+            sage: I = K.fractional_ideal((2, -7/44*a^5 + 19/44*a^4 + 87/44*a^3 - 87/44*a^2 - 5/2*a + 39/22))
+            sage: I.norm()
+            2
+            sage: I in K.primes_above(2)
+            True
+            sage: K.valuation(I)
+            [ 2-adic valuation, v(x + 1) = 1/2 ]-adic valuation
+
+        ::
+
+            sage: K.<a, b> = NumberField([x^2 - 2, x^2 + x + 1])
+            sage: K.valuation(2)
+            2-adic valuation
+
         """
         K, L, G = self._normalize_number_field_data(R)
 
@@ -297,14 +317,33 @@ def create_key_and_extra_args_for_number_field_from_ideal(self, R, I, prime):
         if len(F) != 1:
             raise ValueError("%r does not lie over a single prime of %r"%(I, K))
         vK = K.valuation(F[0][0])
-        candidates = vK.mac_lane_approximants(G, require_incomparability=True)
+        approximants = vK.mac_lane_approximants(G, require_incomparability=True)
 
-        candidates_for_I = [c for c in candidates if all(c(g.polynomial()) > 0 for g in I.gens())]
-        assert(len(candidates_for_I) > 0) # This should not be possible, unless I contains a unit
-        if len(candidates_for_I) > 1:
-            raise ValueError("%s does not single out a unique extension of %s to %s"%(prime, vK, L))
-        else:
-            return (R, candidates_for_I[0]), {'approximants': candidates}
+        candidates = approximants[:]
+
+        # The correct approximant has v(g) > 0 for all g in the ideal.
+        # Unfortunately, the generators of I, even though defined over K have
+        # their polynomial() defined over the rationals so we need to turn them
+        # into polynomials over K[x] explicitly.
+        from sage.rings.all import PolynomialRing
+        gens = I.gens()
+        gens = [PolynomialRing(K, 'x')(list(g.vector())) for g in gens]
+
+        # Refine candidates until we can detect which valuation corresponds to the ideal I
+        while True:
+            assert any(candidates), "the defining polynomial of the extension factored but we still could not figure out which valuation corresponds to the given ideal"
+
+            match = [i for (i, v) in enumerate(candidates) if v and all(v(g) > 0 for g in gens)]
+
+            if len(match) > 1:
+                raise ValueError("%s does not single out a unique extension of %s to %s"%(prime, vK, L))
+            if len(match) == 1:
+                return (R, approximants[match[0]]), {'approximants': approximants}
+
+            # We refine candidates which increases v(g) for all g in I;
+            # however, we cannot augment the valuations which are already at
+            # v(G) = +∞ which we ignore by setting them to None.
+            candidates = [v.mac_lane_step(G)[0] if v and v.is_discrete_valuation() else None for v in candidates]
 
     def _normalize_number_field_data(self, R):
         r"""