Fixing some documentation, moving code up to sage.groups.galois_group

src/sage/groups/galois_group.py

@@ -1,4 +1,4 @@
-"""
+r"""
 Galois groups of field extensions
 
 AUTHORS:
@@ -15,7 +15,7 @@
 from functools import wraps
 
 def _alg_key(self, algorithm=None, recompute=False):
-    """
+    r"""
     Return a key for use in cached_method calls.
 
     If recompute is false, will cache using ``None`` as the key, so no recomputation will be done.
@@ -36,7 +36,7 @@ def _alg_key(self, algorithm=None, recompute=False):
         return algorithm
 
 class GaloisGroup(PermutationGroup_generic):
-    """
+    r"""
     The group of automorphisms of a Galois closure of a given field.
 
     INPUT:
@@ -50,6 +50,14 @@ class GaloisGroup(PermutationGroup_generic):
         of the original extension).  The default value may vary based on the type of field.
     """
     def __init__(self, field, algorithm=None, names=None, gc_numbering=False):
+        r"""
+        EXAMPLES::
+
+            sage: R.<x> = ZZ[]
+            sage: K.<a> = NumberField(x^3 + 2*x + 2)
+            sage: G = K.galois_group()
+            sage: TestSuite(G).run()
+        """
         self._field = field
         self._default_algorithm = algorithm
         self._base = field.base_field()
@@ -66,8 +74,24 @@ def __init__(self, field, algorithm=None, names=None, gc_numbering=False):
         # Instead, the relevant attributes are computed lazily
         super(PermutationGroup_generic, self).__init__(category=category)
 
-    def _get_algorithm(self, algorithm):
+    def _repr_(self):
         """
+        String representation of this Galois group
+
+        EXAMPLES::
+
+            sage: from sage.groups.galois_group import GaloisGroup
+            sage: R.<x> = ZZ[]
+            sage: K.<a> = NumberField(x^3 + 2*x + 2)
+            sage: G = K.galois_group()
+            sage: GaloisGroup._repr_(G)
+            'Galois group of x^3 + 2*x + 2'
+        """
+        f = self._field.defining_polynomial()
+        return "Galois group of %s" % f
+
+    def _get_algorithm(self, algorithm):
+        r"""
         Allows overriding the default algorithm specified at object creation.
 
         EXAMPLES::
@@ -87,13 +111,15 @@ def _get_algorithm(self, algorithm):
     # methods (taking algorithm and recompute as arguments):
     # * transitive_number
     # * order
+    # * _element_constructor_ -- for creating elements
 
     # lazy_attributes
     # * _gcdata -- a pair, the Galois closure and an embedding of the top field into it
-
+    # * _gens -- the list of generators of this group, as elements.  This is not computed during __init__ for speed
+    # * _elts -- the list of all elements of this group.
 
     def top_field(self):
-        """
+        r"""
         Return the larger of the two fields in the extension defining this Galois group.
 
         Note that this field may not be Galois.
@@ -113,7 +139,7 @@ def top_field(self):
         return self._field
 
     def transitive_label(self):
-        """
+        r"""
         Return the transitive label for the action of this Galois group on the roots of
         the defining polynomial of the field extension.
 
@@ -125,11 +151,29 @@ def transitive_label(self):
             sage: G.transitive_label()
             '8T44'
         """
-        return "%sT%s" % (self._field.degree(), self.transitive_number())
+        try:
+            return "%sT%s" % (self._field.degree(), self.transitive_number())
+        except NotImplementedError: # relative number fields don't support degree
+            return "%sT%s" % (self._field.relative_degree(), self.transitive_number())
+
+    def is_galois(self):
+        r"""
+        Return whether the top field is Galois over its base.
+
+        EXAMPLES::
+
+            sage: R.<x> = ZZ[]
+            sage: K.<a> = NumberField(x^8 - x^5 + x^4 - x^3 + 1)
+            sage: G = K.galois_group()
+            sage: from sage.groups.galois_group import GaloisGroup
+            sage: GaloisGroup.is_galois(G)
+            False
+        """
+        return self.order() == self._field.degree()
 
     @lazy_attribute
     def _galois_closure(self):
-        """
+        r"""
         The Galois closure of the top field.
 
         EXAMPLES::
@@ -142,9 +186,23 @@ def _galois_closure(self):
         """
         return self._gcdata[0]
 
+    def splitting_field(self):
+        r"""
+        The Galois closure of the top field.
+
+        EXAMPLES::
+
+            sage: K = NumberField(x^3 - x + 1, 'a')
+            sage: K.galois_group(names='b').splitting_field()
+            Number Field in b with defining polynomial x^6 - 6*x^4 + 9*x^2 + 23
+            sage: L = QuadraticField(-23, 'c'); L.galois_group().splitting_field() is L
+            True
+        """
+        return self._galois_closure
+
     @lazy_attribute
     def _gc_map(self):
-        """
+        r"""
         The inclusion of the top field into the Galois closure.
 
         EXAMPLES::
@@ -162,7 +220,7 @@ def _gc_map(self):
 
     @lazy_attribute
     def _deg(self):
-        """
+        r"""
         The number of moved points in the permutation representation.
 
         This will be the degree of the original number field if `_gc_numbering``
@@ -182,11 +240,14 @@ def _deg(self):
         if self._gc_numbering:
             return self.order()
         else:
-            return self._field.degree()
+            try:
+                return self._field.degree()
+            except NotImplementedError: # relative number fields don't support degree
+                return self._field.relative_degree()
 
     @lazy_attribute
     def _domain(self):
-        """
+        r"""
         The integers labeling the roots on which this Galois group acts.
 
         EXAMPLES::
@@ -204,7 +265,7 @@ def _domain(self):
 
     @lazy_attribute
     def _domain_to_gap(self):
-        """
+        r"""
         Dictionary implementing the identity (used by PermutationGroup_generic).
 
         EXAMPLES::
@@ -219,7 +280,7 @@ def _domain_to_gap(self):
 
     @lazy_attribute
     def _domain_from_gap(self):
-        """
+        r"""
         Dictionary implementing the identity (used by PermutationGroup_generic).
 
         EXAMPLES::
@@ -232,3 +293,13 @@ def _domain_from_gap(self):
         """
         return dict((i+1, key) for i, key in enumerate(self._domain))
 
+    def ngens(self):
+        r"""
+        Number of generators of this Galois group
+
+        EXAMPLES::
+
+            sage: QuadraticField(-23, 'a').galois_group().ngens()
+            1
+        """
+        return len(self._gens)

src/sage/rings/number_field/galois_group.py

@@ -601,17 +601,6 @@ def is_galois(self):
         else:
             return len(K.automorphisms()) == d
 
-    def ngens(self):
-        r"""
-        Number of generators of this Galois group
-
-        EXAMPLES::
-
-            sage: QuadraticField(-23, 'a').galois_group().ngens()
-            1
-        """
-        return len(self._gens)
-
     def _repr_(self):
         r"""
         String representation of this Galois group
@@ -650,20 +639,6 @@ def number_field(self):
         """
         return self._field
 
-    def splitting_field(self):
-        r"""
-        The Galois closure of the ambient number field.
-
-        EXAMPLES::
-
-            sage: K = NumberField(x^3 - x + 1, 'a')
-            sage: K.galois_group(names='b').splitting_field()
-            Number Field in b with defining polynomial x^6 - 6*x^4 + 9*x^2 + 23
-            sage: L = QuadraticField(-23, 'c'); L.galois_group().splitting_field() is L
-            True
-        """
-        return self._galois_closure
-
     def list(self):
         r"""
         List of the elements of self.