Base class for generic \(p\)-adic polynomials¶
This provides common functionality for all \(p\)-adic polynomials, such as printing and factoring.
AUTHORS:
Jeroen Demeyer (2013-11-22): initial version, split off from other files, made Polynomial_padic the common base class for all p-adic polynomials.
- class sage.rings.polynomial.padics.polynomial_padic.Polynomial_padic(parent, x=None, check=True, is_gen=False, construct=False)[source]¶
Bases:
Polynomial- content()[source]¶
Compute the content of this polynomial.
OUTPUT:
If this is the zero polynomial, return the constant coefficient. Otherwise, since the content is only defined up to a unit, return the content as \(\pi^k\) with maximal precision where \(k\) is the minimal valuation of any of the coefficients.
EXAMPLES:
sage: # needs sage.libs.ntl sage: K = Zp(13,7) sage: R.<t> = K[] sage: f = 13^7*t^3 + K(169,4)*t - 13^4 sage: f.content() 13^2 + O(13^9) sage: R(0).content() 0 sage: f = R(K(0,3)); f O(13^3) sage: f.content() O(13^3) sage: # needs sage.libs.ntl sage: P.<x> = ZZ[] sage: f = x + 2 sage: f.content() 1 sage: fp = f.change_ring(pAdicRing(2, 10)) sage: fp (1 + O(2^10))*x + 2 + O(2^11) sage: fp.content() 1 + O(2^10) sage: (2*fp).content() 2 + O(2^11)
>>> from sage.all import * >>> # needs sage.libs.ntl >>> K = Zp(Integer(13),Integer(7)) >>> R = K['t']; (t,) = R._first_ngens(1) >>> f = Integer(13)**Integer(7)*t**Integer(3) + K(Integer(169),Integer(4))*t - Integer(13)**Integer(4) >>> f.content() 13^2 + O(13^9) >>> R(Integer(0)).content() 0 >>> f = R(K(Integer(0),Integer(3))); f O(13^3) >>> f.content() O(13^3) >>> # needs sage.libs.ntl >>> P = ZZ['x']; (x,) = P._first_ngens(1) >>> f = x + Integer(2) >>> f.content() 1 >>> fp = f.change_ring(pAdicRing(Integer(2), Integer(10))) >>> fp (1 + O(2^10))*x + 2 + O(2^11) >>> fp.content() 1 + O(2^10) >>> (Integer(2)*fp).content() 2 + O(2^11)
# needs sage.libs.ntl K = Zp(13,7) R.<t> = K[] f = 13^7*t^3 + K(169,4)*t - 13^4 f.content() R(0).content() f = R(K(0,3)); f f.content() # needs sage.libs.ntl P.<x> = ZZ[] f = x + 2 f.content() fp = f.change_ring(pAdicRing(2, 10)) fp fp.content() (2*fp).content()
Over a field it would be sufficient to return only zero or one, as the content is only defined up to multiplication with a unit. However, we return \(\pi^k\) where \(k\) is the minimal valuation of any coefficient:
sage: # needs sage.libs.ntl sage: K = Qp(13,7) sage: R.<t> = K[] sage: f = 13^7*t^3 + K(169,4)*t - 13^-4 sage: f.content() 13^-4 + O(13^3) sage: f = R.zero() sage: f.content() 0 sage: f = R(K(0,3)) sage: f.content() O(13^3) sage: f = 13*t^3 + K(0,1)*t sage: f.content() 13 + O(13^8)
>>> from sage.all import * >>> # needs sage.libs.ntl >>> K = Qp(Integer(13),Integer(7)) >>> R = K['t']; (t,) = R._first_ngens(1) >>> f = Integer(13)**Integer(7)*t**Integer(3) + K(Integer(169),Integer(4))*t - Integer(13)**-Integer(4) >>> f.content() 13^-4 + O(13^3) >>> f = R.zero() >>> f.content() 0 >>> f = R(K(Integer(0),Integer(3))) >>> f.content() O(13^3) >>> f = Integer(13)*t**Integer(3) + K(Integer(0),Integer(1))*t >>> f.content() 13 + O(13^8)
# needs sage.libs.ntl K = Qp(13,7) R.<t> = K[] f = 13^7*t^3 + K(169,4)*t - 13^-4 f.content() f = R.zero() f.content() f = R(K(0,3)) f.content() f = 13*t^3 + K(0,1)*t f.content()
- factor()[source]¶
Return the factorization of this polynomial.
EXAMPLES:
sage: # needs sage.libs.ntl sage: R.<t> = PolynomialRing(Qp(3, 3, print_mode='terse', print_pos=False)) sage: pol = t^8 - 1 sage: for p,e in pol.factor(): ....: print("{} {}".format(e, p)) 1 (1 + O(3^3))*t + 1 + O(3^3) 1 (1 + O(3^3))*t - 1 + O(3^3) 1 (1 + O(3^3))*t^2 + (5 + O(3^3))*t - 1 + O(3^3) 1 (1 + O(3^3))*t^2 + (-5 + O(3^3))*t - 1 + O(3^3) 1 (1 + O(3^3))*t^2 + O(3^3)*t + 1 + O(3^3) sage: R.<t> = PolynomialRing(Qp(5, 6, print_mode='terse', print_pos=False)) sage: pol = 100 * (5*t - 1) * (t - 5); pol (500 + O(5^9))*t^2 + (-2600 + O(5^8))*t + 500 + O(5^9) sage: pol.factor() (500 + O(5^9)) * ((1 + O(5^5))*t - 1/5 + O(5^5)) * ((1 + O(5^6))*t - 5 + O(5^6)) sage: pol.factor().value() (500 + O(5^8))*t^2 + (-2600 + O(5^8))*t + 500 + O(5^8)
>>> from sage.all import * >>> # needs sage.libs.ntl >>> R = PolynomialRing(Qp(Integer(3), Integer(3), print_mode='terse', print_pos=False), names=('t',)); (t,) = R._first_ngens(1) >>> pol = t**Integer(8) - Integer(1) >>> for p,e in pol.factor(): ... print("{} {}".format(e, p)) 1 (1 + O(3^3))*t + 1 + O(3^3) 1 (1 + O(3^3))*t - 1 + O(3^3) 1 (1 + O(3^3))*t^2 + (5 + O(3^3))*t - 1 + O(3^3) 1 (1 + O(3^3))*t^2 + (-5 + O(3^3))*t - 1 + O(3^3) 1 (1 + O(3^3))*t^2 + O(3^3)*t + 1 + O(3^3) >>> R = PolynomialRing(Qp(Integer(5), Integer(6), print_mode='terse', print_pos=False), names=('t',)); (t,) = R._first_ngens(1) >>> pol = Integer(100) * (Integer(5)*t - Integer(1)) * (t - Integer(5)); pol (500 + O(5^9))*t^2 + (-2600 + O(5^8))*t + 500 + O(5^9) >>> pol.factor() (500 + O(5^9)) * ((1 + O(5^5))*t - 1/5 + O(5^5)) * ((1 + O(5^6))*t - 5 + O(5^6)) >>> pol.factor().value() (500 + O(5^8))*t^2 + (-2600 + O(5^8))*t + 500 + O(5^8)
# needs sage.libs.ntl R.<t> = PolynomialRing(Qp(3, 3, print_mode='terse', print_pos=False)) pol = t^8 - 1 for p,e in pol.factor(): print("{} {}".format(e, p)) R.<t> = PolynomialRing(Qp(5, 6, print_mode='terse', print_pos=False)) pol = 100 * (5*t - 1) * (t - 5); pol pol.factor() pol.factor().value()The same factorization over \(\ZZ_p\). In this case, the “unit” part is a \(p\)-adic unit and the power of \(p\) is considered to be a factor:
sage: # needs sage.libs.ntl sage: R.<t> = PolynomialRing(Zp(5, 6, print_mode='terse', print_pos=False)) sage: pol = 100 * (5*t - 1) * (t - 5); pol (500 + O(5^9))*t^2 + (-2600 + O(5^8))*t + 500 + O(5^9) sage: pol.factor() (4 + O(5^6)) * (5 + O(5^7))^2 * ((1 + O(5^6))*t - 5 + O(5^6)) * ((5 + O(5^6))*t - 1 + O(5^6)) sage: pol.factor().value() (500 + O(5^8))*t^2 + (-2600 + O(5^8))*t + 500 + O(5^8)
>>> from sage.all import * >>> # needs sage.libs.ntl >>> R = PolynomialRing(Zp(Integer(5), Integer(6), print_mode='terse', print_pos=False), names=('t',)); (t,) = R._first_ngens(1) >>> pol = Integer(100) * (Integer(5)*t - Integer(1)) * (t - Integer(5)); pol (500 + O(5^9))*t^2 + (-2600 + O(5^8))*t + 500 + O(5^9) >>> pol.factor() (4 + O(5^6)) * (5 + O(5^7))^2 * ((1 + O(5^6))*t - 5 + O(5^6)) * ((5 + O(5^6))*t - 1 + O(5^6)) >>> pol.factor().value() (500 + O(5^8))*t^2 + (-2600 + O(5^8))*t + 500 + O(5^8)
# needs sage.libs.ntl R.<t> = PolynomialRing(Zp(5, 6, print_mode='terse', print_pos=False)) pol = 100 * (5*t - 1) * (t - 5); pol pol.factor() pol.factor().value()
In the following example, the discriminant is zero, so the \(p\)-adic factorization is not well defined:
sage: factor(t^2) # needs sage.libs.ntl Traceback (most recent call last): ... PrecisionError: p-adic factorization not well-defined since the discriminant is zero up to the requestion p-adic precision
>>> from sage.all import * >>> factor(t**Integer(2)) # needs sage.libs.ntl Traceback (most recent call last): ... PrecisionError: p-adic factorization not well-defined since the discriminant is zero up to the requestion p-adic precision
factor(t^2) # needs sage.libs.ntl
An example of factoring a constant polynomial (see Issue #26669):
sage: R.<x> = Qp(5)[] # needs sage.libs.ntl sage: R(2).factor() # needs sage.libs.ntl 2 + O(5^20)
>>> from sage.all import * >>> R = Qp(Integer(5))['x']; (x,) = R._first_ngens(1)# needs sage.libs.ntl >>> R(Integer(2)).factor() # needs sage.libs.ntl 2 + O(5^20)
R.<x> = Qp(5)[] # needs sage.libs.ntl R(2).factor() # needs sage.libs.ntl
More examples over \(\ZZ_p\):
sage: R.<w> = PolynomialRing(Zp(5, prec=6, type='capped-abs', print_mode='val-unit')) sage: f = w^5 - 1 sage: f.factor() # needs sage.libs.ntl ((1 + O(5^6))*w + 3124 + O(5^6)) * ((1 + O(5^6))*w^4 + (12501 + O(5^6))*w^3 + (9376 + O(5^6))*w^2 + (6251 + O(5^6))*w + 3126 + O(5^6))
>>> from sage.all import * >>> R = PolynomialRing(Zp(Integer(5), prec=Integer(6), type='capped-abs', print_mode='val-unit'), names=('w',)); (w,) = R._first_ngens(1) >>> f = w**Integer(5) - Integer(1) >>> f.factor() # needs sage.libs.ntl ((1 + O(5^6))*w + 3124 + O(5^6)) * ((1 + O(5^6))*w^4 + (12501 + O(5^6))*w^3 + (9376 + O(5^6))*w^2 + (6251 + O(5^6))*w + 3126 + O(5^6))
R.<w> = PolynomialRing(Zp(5, prec=6, type='capped-abs', print_mode='val-unit')) f = w^5 - 1 f.factor() # needs sage.libs.ntl
See Issue #4038:
sage: # needs sage.libs.ntl sage.schemes sage: E = EllipticCurve('37a1') sage: K = Qp(7,10) sage: EK = E.base_extend(K) sage: g = EK.division_polynomial_0(3) sage: g.factor() (3 + O(7^10)) * ((1 + O(7^10))*x + 1 + 2*7 + 4*7^2 + 2*7^3 + 5*7^4 + 7^5 + 5*7^6 + 3*7^7 + 5*7^8 + 3*7^9 + O(7^10)) * ((1 + O(7^10))*x^3 + (6 + 4*7 + 2*7^2 + 4*7^3 + 7^4 + 5*7^5 + 7^6 + 3*7^7 + 7^8 + 3*7^9 + O(7^10))*x^2 + (6 + 3*7 + 5*7^2 + 2*7^4 + 7^5 + 7^6 + 2*7^8 + 3*7^9 + O(7^10))*x + 2 + 5*7 + 4*7^2 + 2*7^3 + 6*7^4 + 3*7^5 + 7^6 + 4*7^7 + O(7^10))
>>> from sage.all import * >>> # needs sage.libs.ntl sage.schemes >>> E = EllipticCurve('37a1') >>> K = Qp(Integer(7),Integer(10)) >>> EK = E.base_extend(K) >>> g = EK.division_polynomial_0(Integer(3)) >>> g.factor() (3 + O(7^10)) * ((1 + O(7^10))*x + 1 + 2*7 + 4*7^2 + 2*7^3 + 5*7^4 + 7^5 + 5*7^6 + 3*7^7 + 5*7^8 + 3*7^9 + O(7^10)) * ((1 + O(7^10))*x^3 + (6 + 4*7 + 2*7^2 + 4*7^3 + 7^4 + 5*7^5 + 7^6 + 3*7^7 + 7^8 + 3*7^9 + O(7^10))*x^2 + (6 + 3*7 + 5*7^2 + 2*7^4 + 7^5 + 7^6 + 2*7^8 + 3*7^9 + O(7^10))*x + 2 + 5*7 + 4*7^2 + 2*7^3 + 6*7^4 + 3*7^5 + 7^6 + 4*7^7 + O(7^10))
# needs sage.libs.ntl sage.schemes E = EllipticCurve('37a1') K = Qp(7,10) EK = E.base_extend(K) g = EK.division_polynomial_0(3) g.factor()
- root_field(names, check_irreducible=True, **kwds)[source]¶
Return the \(p\)-adic extension field generated by the roots of the irreducible polynomial
self.INPUT:
names– name of the generator of the extensioncheck_irreducible– check whether the polynomial is irreduciblekwds– seesage.rings.padics.padic_generic.pAdicGeneric.extension()
EXAMPLES:
sage: R.<x> = Qp(3,5,print_mode='digits')[] # needs sage.libs.ntl sage: f = x^2 - 3 # needs sage.libs.ntl sage: f.root_field('x') # needs sage.libs.ntl 3-adic Eisenstein Extension Field in x defined by x^2 - 3
>>> from sage.all import * >>> R = Qp(Integer(3),Integer(5),print_mode='digits')['x']; (x,) = R._first_ngens(1)# needs sage.libs.ntl >>> f = x**Integer(2) - Integer(3) # needs sage.libs.ntl >>> f.root_field('x') # needs sage.libs.ntl 3-adic Eisenstein Extension Field in x defined by x^2 - 3
R.<x> = Qp(3,5,print_mode='digits')[] # needs sage.libs.ntl f = x^2 - 3 # needs sage.libs.ntl f.root_field('x') # needs sage.libs.ntlsage: R.<x> = Qp(5,5,print_mode='digits')[] # needs sage.libs.ntl sage: f = x^2 - 3 # needs sage.libs.ntl sage: f.root_field('x', print_mode='bars') # needs sage.libs.ntl 5-adic Unramified Extension Field in x defined by x^2 - 3
>>> from sage.all import * >>> R = Qp(Integer(5),Integer(5),print_mode='digits')['x']; (x,) = R._first_ngens(1)# needs sage.libs.ntl >>> f = x**Integer(2) - Integer(3) # needs sage.libs.ntl >>> f.root_field('x', print_mode='bars') # needs sage.libs.ntl 5-adic Unramified Extension Field in x defined by x^2 - 3
R.<x> = Qp(5,5,print_mode='digits')[] # needs sage.libs.ntl f = x^2 - 3 # needs sage.libs.ntl f.root_field('x', print_mode='bars') # needs sage.libs.ntlsage: R.<x> = Qp(11,5,print_mode='digits')[] # needs sage.libs.ntl sage: f = x^2 - 3 # needs sage.libs.ntl sage: f.root_field('x', print_mode='bars') # needs sage.libs.ntl Traceback (most recent call last): ... ValueError: polynomial must be irreducible
>>> from sage.all import * >>> R = Qp(Integer(11),Integer(5),print_mode='digits')['x']; (x,) = R._first_ngens(1)# needs sage.libs.ntl >>> f = x**Integer(2) - Integer(3) # needs sage.libs.ntl >>> f.root_field('x', print_mode='bars') # needs sage.libs.ntl Traceback (most recent call last): ... ValueError: polynomial must be irreducible
R.<x> = Qp(11,5,print_mode='digits')[] # needs sage.libs.ntl f = x^2 - 3 # needs sage.libs.ntl f.root_field('x', print_mode='bars') # needs sage.libs.ntl