34212: clean version

Author: guojing0

src/sage/rings/number_field/number_field.py

@@ -153,6 +153,10 @@
 from collections import Counter
 from builtins import zip
 
+from sage.categories.homset import Hom
+from sage.categories.sets_cat import Sets
+from sage.modules.free_module import VectorSpace
+from sage.modules.free_module_element import vector
 
 _NumberFields = NumberFields()
 
@@ -9297,6 +9301,69 @@ def minkowski_embedding(self, B=None, prec=None):
 
         return sage.matrix.all.matrix(d)
 
+    def logarithmic_embedding(self, prec=53):
+        """
+        Return the morphism of ``self`` under the logarithmic embedding
+        in the category Set.
+
+        The logarithmic embedding is defined as a map from the number field ``self`` to `\RR^n`.
+
+        It is defined under Definition 4.9.6 in [Cohen1993]_.
+
+        INPUT:
+
+        - ``prec`` -- desired floating point precision.
+
+        OUTPUT:
+
+        - a tuple of real numbers.
+
+        EXAMPLES::
+
+            sage: CF.<a> = CyclotomicField(97)
+            sage: f = CF.logarithmic_embedding()
+            sage: f(0)
+            (-1)
+            sage: f(7)
+            (1.94591014905531)
+
+        ::
+
+            sage: K.<a> = NumberField(x^3 + 5)
+            sage: f = K.logarithmic_embedding()
+            sage: f(0)
+            (-1)
+            sage: f(7)
+            (1.94591014905531)
+        """
+        def closure_map(x, prec=53):
+            """
+            The function closure of the logarithmic embedding.
+            """
+            K = self.base_ring()
+            K_embeddings = K.places(prec)
+            r1, r2 = K.signature()
+            r = r1 + r2 - 1
+
+            from sage.rings.all import RealField
+            Reals = RealField(prec)
+
+            if x == 0:
+                return vector([-1 for _ in range(r + 1)])
+
+            x_logs = []
+            for i in range(r1):
+                sigma = K_embeddings[i]
+                x_logs.append(Reals(abs(sigma(x))).log())
+            for i in range(r1, r + 1):
+                tau = K_embeddings[i]
+                x_logs.append(2 * Reals(abs(tau(x))).log())
+
+            return vector(x_logs)
+
+        log_map = closure_map(self(0), prec)
+        return Hom(self, VectorSpace(QQ, len(log_map)), Sets())
+
     def places(self, all_complex=False, prec=None):
         r"""
         Return the collection of all infinite places of self.