def to_number_field(extension, theta=None, **args): """Express `extension` in the field generated by `theta`. """ gen = args.get('gen') if hasattr(extension, '__iter__'): extension = list(extension) else: extension = [extension] if len(extension) == 1 and type(extension[0]) is tuple: return AlgebraicNumber(extension[0]) minpoly, coeffs = primitive_element(extension, gen, polys=True) root = sum([coeff * ext for coeff, ext in zip(coeffs, extension)]) if theta is None: return AlgebraicNumber((minpoly, root)) else: theta = sympify(theta) if not theta.is_AlgebraicNumber: theta = AlgebraicNumber(theta, gen=gen) coeffs = field_isomorphism(root, theta) if coeffs is not None: return AlgebraicNumber(theta, coeffs) else: raise IsomorphismFailed("%s is not in a subfield of %s" % (root, theta.root))
def field_isomorphism(a, b, **args): """Construct an isomorphism between two number fields. """ a, b = sympify(a), sympify(b) if not a.is_AlgebraicNumber: a = AlgebraicNumber(a) if not b.is_AlgebraicNumber: b = AlgebraicNumber(b) if a == b: return a.coeffs() n = a.minpoly.degree() m = b.minpoly.degree() if n == 1: return [a.root] if m % n != 0: return None if args.get('fast', True): try: result = field_isomorphism_pslq(a, b) if result is not None: return result except NotImplementedError: pass return field_isomorphism_factor(a, b)
def field_isomorphism(a, b, **args): """Construct an isomorphism between two number fields. """ a, b = sympify(a), sympify(b) if not a.is_AlgebraicNumber: a = AlgebraicNumber(a) if not b.is_AlgebraicNumber: b = AlgebraicNumber(b) if a == b: return a.coeffs() n = a.minpoly.degree() m = b.minpoly.degree() if n == 1: return [a.root] if m % n != 0: return None if args.get('fast', True): try: result = field_isomorphism_pslq(a, b) if result is not None: return result except NotImplementedError: pass return field_isomorphism_factor(a, b)
def test_AlgebraicNumber(): a = AlgebraicNumber(sqrt(2)) sT( a, "AlgebraicNumber(Pow(Integer(2), Rational(1, 2)), [Integer(1), Integer(0)])" ) a = AlgebraicNumber(root(-2, 3)) sT( a, "AlgebraicNumber(Pow(Integer(-2), Rational(1, 3)), [Integer(1), Integer(0)])" )
def primitive_element(extension, x=None, **args): """Construct a common number field for all extensions. """ if not extension: raise ValueError("can't compute primitive element for empty extension") if x is not None: x, cls = sympify(x), Poly else: x, cls = Dummy('x'), PurePoly if not args.get('ex', False): extension = [AlgebraicNumber(ext, gen=x) for ext in extension] g, coeffs = extension[0].minpoly.replace(x), [1] for ext in extension[1:]: s, _, g = sqf_norm(g, x, extension=ext) coeffs = [s * c for c in coeffs] + [1] if not args.get('polys', False): return g.as_expr(), coeffs else: return cls(g), coeffs generator = numbered_symbols('y', cls=Dummy) F, Y = [], [] for ext in extension: y = next(generator) if ext.is_Poly: if ext.is_univariate: f = ext.as_expr(y) else: raise ValueError("expected minimal polynomial, got %s" % ext) else: f = minpoly(ext, y) F.append(f) Y.append(y) coeffs_generator = args.get('coeffs', _coeffs_generator) for coeffs in coeffs_generator(len(Y)): f = x - sum([c * y for c, y in zip(coeffs, Y)]) G = groebner(F + [f], Y + [x], order='lex', field=True) H, g = G[:-1], cls(G[-1], x, domain='QQ') for i, (h, y) in enumerate(zip(H, Y)): try: H[i] = Poly(y - h, x, domain='QQ').all_coeffs() # XXX: composite=False except CoercionFailed: # pragma: no cover break # G is not a triangular set else: break else: # pragma: no cover raise RuntimeError("run out of coefficient configurations") _, g = g.clear_denoms() if not args.get('polys', False): return g.as_expr(), coeffs, H else: return g, coeffs, H
def test_H24(): phi = AlgebraicNumber(S.GoldenRatio.expand(func=True), alias='phi') assert factor(x**4 - 3*x**2 + 1, extension=phi) == \ (x - phi)*(x + 1 - phi)*(x - 1 + phi)*(x + phi)