Added _factor_univariate_polynomial() for QQbar and AA

Author: pjbruin

src/sage/rings/polynomial/polynomial_element.pyx

@@ -3073,27 +3073,13 @@ cdef class Polynomial(CommutativeAlgebraElement):
             sage: R.<x> = QQbar[]
             sage: (x^8-1).factor()
             (x - 1) * (x - 0.7071067811865475? - 0.7071067811865475?*I) * (x - 0.7071067811865475? + 0.7071067811865475?*I) * (x - I) * (x + I) * (x + 0.7071067811865475? - 0.7071067811865475?*I) * (x + 0.7071067811865475? + 0.7071067811865475?*I) * (x + 1)
-            sage: (12*x^2-4).factor()
-            (12) * (x - 0.5773502691896258?) * (x + 0.5773502691896258?)
-            sage: R(-1).factor()
-            -1
-            sage: EllipticCurve('11a1').change_ring(QQbar).division_polynomial(5).factor()
-            (5) * (x - 16) * (x - 5) * (x - 1.959674775249769?) * (x - 1.427050983124843? - 3.665468789467727?*I) * (x - 1.427050983124843? + 3.665468789467727?*I) * (x + 0.9549150281252629? - 0.8652998037182486?*I) * (x + 0.9549150281252629? + 0.8652998037182486?*I) * (x + 1.927050983124843? - 1.677599044300515?*I) * (x + 1.927050983124843? + 1.677599044300515?*I) * (x + 2.959674775249769?) * (x + 6.545084971874737? - 7.106423590645660?*I) * (x + 6.545084971874737? + 7.106423590645660?*I)
 
         Factoring polynomials over the algebraic reals (see
         :trac:`8544`)::
 
             sage: R.<x> = AA[]
             sage: (x^8+1).factor()
             (x^2 - 1.847759065022574?*x + 1.000000000000000?) * (x^2 - 0.7653668647301795?*x + 1.000000000000000?) * (x^2 + 0.7653668647301795?*x + 1.000000000000000?) * (x^2 + 1.847759065022574?*x + 1.000000000000000?)
-            sage: R(3).factor()
-            3
-            sage: (12*x^2-4).factor()
-            (12) * (x - 0.5773502691896258?) * (x + 0.5773502691896258?)
-            sage: (12*x^2+4).factor()
-            (12) * (x^2 + 0.3333333333333334?)
-            sage: EllipticCurve('11a1').change_ring(AA).division_polynomial(5).factor()
-            (5) * (x - 16.00000000000000?) * (x - 5.000000000000000?) * (x - 1.959674775249769?) * (x + 2.959674775249769?) * (x^2 - 2.854101966249685?*x + 15.47213595499958?) * (x^2 + 1.909830056250526?*x + 1.660606461254312?) * (x^2 + 3.854101966249685?*x + 6.527864045000421?) * (x^2 + 13.09016994374948?*x + 93.33939353874569?)
 
         TESTS:
 
@@ -3350,21 +3336,6 @@ cdef class Polynomial(CommutativeAlgebraElement):
             f = pari(v).Polrev()
             G = list(f.factor())
 
-        elif is_AlgebraicField(R):
-            Rx = self.parent()
-            return Factorization([(Rx([-r,1]),e) for r,e in self.roots()],
-                                 unit=self.leading_coefficient())
-
-        elif is_AlgebraicRealField(R):
-            from sage.rings.qqbar import QQbar
-            Rx = self.parent()
-            rr = self.roots()
-            cr = [(r,e) for r,e in self.roots(QQbar) if r.imag()>0]
-            return Factorization(
-                [(Rx([-r,1]),e) for r,e in rr] +
-                [(Rx([r.norm(),-2*r.real(),1]),e) for r,e in cr],
-                unit=self.leading_coefficient())
-
         elif is_NumberField(R):
             if R.degree() == 1:
                 factors = self.change_ring(QQ).factor()

src/sage/rings/qqbar.py

@@ -959,6 +959,54 @@ def polynomial_root(self, poly, interval, multiplicity=1):
 
         return AlgebraicReal(ANRoot(poly, interval, multiplicity))
 
+    def _factor_univariate_polynomial(self, f):
+        """
+        Factor the univariate polynomial ``f``.
+
+        INPUT:
+
+        - ``f`` -- a univariate polynomial defined over the real algebraic field
+
+        OUTPUT:
+
+        - A factorization of ``f`` over the real algebraic numbers into a unit
+          and monic irreducible factors
+
+        .. NOTE::
+
+            This is a helper method for
+            :meth:`sage.rings.polynomial.polynomial_element.Polynomial.factor`.
+
+        TESTS::
+
+            sage: R.<x> = AA[]
+            sage: AA._factor_univariate_polynomial(x)
+            x
+            sage: AA._factor_univariate_polynomial(2*x)
+            (2) * x
+            sage: AA._factor_univariate_polynomial((x^2 + 1)^2)
+            (x^2 + 1)^2
+            sage: AA._factor_univariate_polynomial(x^8 + 1)
+            (x^2 - 1.847759065022574?*x + 1.000000000000000?) * (x^2 - 0.7653668647301795?*x + 1.000000000000000?) * (x^2 + 0.7653668647301795?*x + 1.000000000000000?) * (x^2 + 1.847759065022574?*x + 1.000000000000000?)
+            sage: AA._factor_univariate_polynomial(R(3))
+            3
+            sage: AA._factor_univariate_polynomial(12*x^2 - 4)
+            (12) * (x - 0.5773502691896258?) * (x + 0.5773502691896258?)
+            sage: AA._factor_univariate_polynomial(12*x^2 + 4)
+            (12) * (x^2 + 0.3333333333333334?)
+            sage: AA._factor_univariate_polynomial(EllipticCurve('11a1').change_ring(AA).division_polynomial(5))
+            (5) * (x - 16.00000000000000?) * (x - 5.000000000000000?) * (x - 1.959674775249769?) * (x + 2.959674775249769?) * (x^2 - 2.854101966249685?*x + 15.47213595499958?) * (x^2 + 1.909830056250526?*x + 1.660606461254312?) * (x^2 + 3.854101966249685?*x + 6.527864045000421?) * (x^2 + 13.09016994374948?*x + 93.33939353874569?)
+
+        """
+        rr = f.roots()
+        cr = [(r,e) for r,e in f.roots(QQbar) if r.imag()>0]
+
+        from sage.structure.factorization import Factorization
+        return Factorization(
+            [(f.parent()([-r,1]),e) for r,e in rr] +
+            [(f.parent()([r.norm(),-2*r.real(),1]),e) for r,e in cr],
+            unit=f.leading_coefficient())
+
 def is_AlgebraicRealField(F):
     r"""
     Check whether ``F`` is an :class:`~AlgebraicRealField` instance. For internal use.
@@ -1355,6 +1403,46 @@ def random_element(self, poly_degree=2, *args, **kwds):
         m = sage.misc.prandom.randint(0, len(roots)-1)
         return roots[m]
 
+    def _factor_univariate_polynomial(self, f):
+        """
+        Factor the univariate polynomial ``f``.
+
+        INPUT:
+
+        - ``f`` -- a univariate polynomial defined over the algebraic field
+
+        OUTPUT:
+
+        - A factorization of ``f`` over the algebraic numbers into a unit and
+          monic irreducible factors
+
+        .. NOTE::
+
+            This is a helper method for
+            :meth:`sage.rings.polynomial.polynomial_element.Polynomial.factor`.
+
+        TESTS::
+
+            sage: R.<x> = QQbar[]
+            sage: QQbar._factor_univariate_polynomial(x)
+            x
+            sage: QQbar._factor_univariate_polynomial(2*x)
+            (2) * x
+            sage: QQbar._factor_univariate_polynomial((x^2 + 1)^2)
+            (x - I)^2 * (x + I)^2
+            sage: QQbar._factor_univariate_polynomial(x^8 - 1)
+            (x - 1) * (x - 0.7071067811865475? - 0.7071067811865475?*I) * (x - 0.7071067811865475? + 0.7071067811865475?*I) * (x - I) * (x + I) * (x + 0.7071067811865475? - 0.7071067811865475?*I) * (x + 0.7071067811865475? + 0.7071067811865475?*I) * (x + 1)
+            sage: QQbar._factor_univariate_polynomial(12*x^2 - 4)
+            (12) * (x - 0.5773502691896258?) * (x + 0.5773502691896258?)
+            sage: QQbar._factor_univariate_polynomial(R(-1))
+            -1
+            sage: QQbar._factor_univariate_polynomial(EllipticCurve('11a1').change_ring(QQbar).division_polynomial(5))
+            (5) * (x - 16) * (x - 5) * (x - 1.959674775249769?) * (x - 1.427050983124843? - 3.665468789467727?*I) * (x - 1.427050983124843? + 3.665468789467727?*I) * (x + 0.9549150281252629? - 0.8652998037182486?*I) * (x + 0.9549150281252629? + 0.8652998037182486?*I) * (x + 1.927050983124843? - 1.677599044300515?*I) * (x + 1.927050983124843? + 1.677599044300515?*I) * (x + 2.959674775249769?) * (x + 6.545084971874737? - 7.106423590645660?*I) * (x + 6.545084971874737? + 7.106423590645660?*I)
+
+        """
+        from sage.structure.factorization import Factorization
+        return Factorization([(f.parent()([-r,1]),e) for r,e in f.roots()], unit=f.leading_coefficient())
+
 def is_AlgebraicField(F):
     r"""
     Check whether ``F`` is an :class:`~AlgebraicField` instance.