34212: log embedding as a morphism

src/sage/rings/number_field/number_field.py

@@ -9293,6 +9293,48 @@ 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: hom = Hom(CF, EuclideanSpace(1), Sets())
+            sage: f = hom(logarithmic_embedding(CF(0)))
+            sage: f(0)
+            (-1)
+            sage: f(7)
+            (1.94591014905531)
+
+        ::
+
+            sage: F.<a> = NumberField(x^3 + 5)
+            sage: K.<b> = F.extension(x^2 + 2)
+            sage: hom = Hom(K, EuclideanSpace(2), Sets())
+            sage: f = hom(logarithmic_embedding(K(0)))
+            sage: f(0)
+            (-1, -1)
+            sage: f(7)
+            (1.94591014905531, 3.89182029811063)
+    """
+    log_map = self._logarithmic_embedding_helper(prec)
+    log_hom = Hom(self.base_ring(), EuclideanSpace(len(log_map(0))), Sets())
+    return log_hom(K(0)._logarithmic_embedding_helper)
+
+def _logarithmic_embedding_helper(self, prec=53):
     """
     Return the image of ``self`` under the logarithmic embedding.