\(p\)-adic Fixed-Mod Element#

Elements of p-adic Rings with Fixed Modulus

AUTHORS:

  • David Roe

  • Genya Zaytman: documentation

  • David Harvey: doctests

class sage.rings.padics.padic_fixed_mod_element.ExpansionIter#

Bases: object

An iterator over a \(p\)-adic expansion.

This class should not be instantiated directly, but instead using expansion().

INPUT:

  • elt – the \(p\)-adic element

  • prec – the number of terms to be emitted

  • mode – either simple_mode, smallest_mode or teichmuller_mode

EXAMPLES:

sage: E = Zp(5,4)(373).expansion()
sage: I = iter(E)  # indirect doctest
sage: type(I)
<class 'sage.rings.padics.padic_capped_relative_element.ExpansionIter'>
class sage.rings.padics.padic_fixed_mod_element.ExpansionIterable#

Bases: object

An iterable storing a \(p\)-adic expansion of an element.

This class should not be instantiated directly, but instead using expansion().

INPUT:

  • elt – the \(p\)-adic element

  • prec – the number of terms to be emitted

  • val_shift – how many zeros to add at the beginning of the expansion, or the number of initial terms to truncate (if negative)

  • mode – one of the following:

    • 'simple_mode'

    • 'smallest_mode'

    • 'teichmuller_mode'

EXAMPLES:

sage: E = Zp(5,4)(373).expansion()  # indirect doctest
sage: type(E)
<class 'sage.rings.padics.padic_capped_relative_element.ExpansionIterable'>
class sage.rings.padics.padic_fixed_mod_element.FMElement#

Bases: pAdicTemplateElement

add_bigoh(absprec)#

Return a new element truncated modulo \(\pi^{\mbox{absprec}}\).

INPUT:

  • absprec – an integer or infinity

OUTPUT:

a new element truncated modulo \(\pi^{\mbox{absprec}}\).

EXAMPLES:

sage: R = Zp(7,4,'fixed-mod','series'); a = R(8); a.add_bigoh(1)
1
is_equal_to(_right, absprec=None)#

Return whether this element is equal to right modulo \(p^{\mbox{absprec}}\).

If absprec is None, returns if self == 0.

INPUT:

  • right – a \(p\)-adic element with the same parent

  • absprec – a positive integer or None (default: None)

EXAMPLES:

sage: R = ZpFM(2, 6)
sage: R(13).is_equal_to(R(13))
True
sage: R(13).is_equal_to(R(13+2^10))
True
sage: R(13).is_equal_to(R(17), 2)
True
sage: R(13).is_equal_to(R(17), 5)
False
is_zero(absprec=None)#

Returns whether self is zero modulo \(\pi^{\mbox{absprec}}\).

INPUT:

  • absprec – an integer

EXAMPLES:

sage: R = ZpFM(17, 6)
sage: R(0).is_zero()
True
sage: R(17^6).is_zero()
True
sage: R(17^2).is_zero(absprec=2)
True
polynomial(var='x')#

Return a polynomial over the base ring that yields this element when evaluated at the generator of the parent.

INPUT:

  • var – string, the variable name for the polynomial

EXAMPLES:

sage: # needs sage.libs.flint
sage: R.<a> = ZqFM(5^3)
sage: a.polynomial()
x
sage: a.polynomial(var='y')
y
sage: (5*a^2 + 25).polynomial()
5*x^2 + 5^2
precision_absolute()#

The absolute precision of this element.

EXAMPLES:

sage: R = Zp(7,4,'fixed-mod'); a = R(7); a.precision_absolute()
4
precision_relative()#

The relative precision of this element.

EXAMPLES:

sage: R = Zp(7,4,'fixed-mod'); a = R(7); a.precision_relative()
3
sage: a = R(0); a.precision_relative()
0
unit_part()#

Return the unit part of self.

If the valuation of self is positive, then the high digits of the result will be zero.

EXAMPLES:

sage: R = Zp(17, 4, 'fixed-mod')
sage: R(5).unit_part()
5
sage: R(18*17).unit_part()
1 + 17
sage: R(0).unit_part()
0
sage: type(R(5).unit_part())
<class 'sage.rings.padics.padic_fixed_mod_element.pAdicFixedModElement'>
sage: R = ZpFM(5, 5); a = R(75); a.unit_part()
3
val_unit()#

Return a 2-tuple, the first element set to the valuation of self, and the second to the unit part of self.

If self == 0, then the unit part is O(p^self.parent().precision_cap()).

EXAMPLES:

sage: R = ZpFM(5,5)
sage: a = R(75); b = a - a
sage: a.val_unit()
(2, 3)
sage: b.val_unit()
(5, 0)
class sage.rings.padics.padic_fixed_mod_element.PowComputer_#

Bases: PowComputer_base

A PowComputer for a fixed-modulus padic ring.

sage.rings.padics.padic_fixed_mod_element.make_pAdicFixedModElement(parent, value)#

Unpickles a fixed modulus element.

EXAMPLES:

sage: from sage.rings.padics.padic_fixed_mod_element import make_pAdicFixedModElement
sage: R = ZpFM(5)
sage: a = make_pAdicFixedModElement(R, 17*25); a
2*5^2 + 3*5^3
class sage.rings.padics.padic_fixed_mod_element.pAdicCoercion_FM_frac_field#

Bases: RingHomomorphism

The canonical inclusion of \(\ZZ_q\) into its fraction field.

EXAMPLES:

sage: # needs sage.libs.flint
sage: R.<a> = ZqFM(27, implementation='FLINT')
sage: K = R.fraction_field()
sage: f = K.coerce_map_from(R); f
Ring morphism:
  From: 3-adic Unramified Extension Ring in a defined by x^3 + 2*x + 1
  To:   3-adic Unramified Extension Field in a defined by x^3 + 2*x + 1
is_injective()#

Return whether this map is injective.

EXAMPLES:

sage: # needs sage.libs.flint
sage: R.<a> = ZqFM(9)
sage: K = R.fraction_field()
sage: f = K.coerce_map_from(R)
sage: f.is_injective()
True
is_surjective()#

Return whether this map is surjective.

EXAMPLES:

sage: # needs sage.libs.flint
sage: R.<a> = ZqFM(9)
sage: K = R.fraction_field()
sage: f = K.coerce_map_from(R)
sage: f.is_surjective()
False
section()#

Return a map back to the ring that converts elements of non-negative valuation.

EXAMPLES:

sage: # needs sage.libs.flint
sage: R.<a> = ZqFM(27)
sage: K = R.fraction_field()
sage: f = K.coerce_map_from(R)
sage: f.section()(K.gen())
a
class sage.rings.padics.padic_fixed_mod_element.pAdicCoercion_ZZ_FM#

Bases: RingHomomorphism

The canonical inclusion from \(\ZZ\) to a fixed modulus ring.

EXAMPLES:

sage: f = ZpFM(5).coerce_map_from(ZZ); f
Ring morphism:
  From: Integer Ring
  To:   5-adic Ring of fixed modulus 5^20
section()#

Returns a map back to \(\ZZ\) that approximates an element of this \(p\)-adic ring by an integer.

EXAMPLES:

sage: f = ZpFM(5).coerce_map_from(ZZ).section()
sage: f(ZpFM(5)(-1)) - 5^20
-1
class sage.rings.padics.padic_fixed_mod_element.pAdicConvert_FM_ZZ#

Bases: RingMap

The map from a fixed modulus ring back to \(\ZZ\) that returns the smallest non-negative integer approximation to its input which is accurate up to the precision.

If the input is not in the closure of the image of \(\ZZ\), raises a ValueError.

EXAMPLES:

sage: f = ZpFM(5).coerce_map_from(ZZ).section(); f
Set-theoretic ring morphism:
  From: 5-adic Ring of fixed modulus 5^20
  To:   Integer Ring
class sage.rings.padics.padic_fixed_mod_element.pAdicConvert_FM_frac_field#

Bases: Morphism

The section of the inclusion from \(\ZZ_q\) to its fraction field.

EXAMPLES:

sage: # needs sage.libs.flint
sage: R.<a> = ZqFM(27)
sage: K = R.fraction_field()
sage: f = R.convert_map_from(K); f
Generic morphism:
  From: 3-adic Unramified Extension Field in a defined by x^3 + 2*x + 1
  To:   3-adic Unramified Extension Ring in a defined by x^3 + 2*x + 1
class sage.rings.padics.padic_fixed_mod_element.pAdicConvert_QQ_FM#

Bases: Morphism

The inclusion map from \(\QQ\) to a fixed modulus ring that is defined on all elements with non-negative \(p\)-adic valuation.

EXAMPLES:

sage: f = ZpFM(5).convert_map_from(QQ); f
Generic morphism:
  From: Rational Field
  To:   5-adic Ring of fixed modulus 5^20
class sage.rings.padics.padic_fixed_mod_element.pAdicFixedModElement#

Bases: FMElement

INPUT:

  • parent – a pAdicRingFixedMod object.

  • x – input data to be converted into the parent.

  • absprec – ignored; for compatibility with other \(p\)-adic rings

  • relprec – ignored; for compatibility with other \(p\)-adic rings

Note

The following types are currently supported for x:

  • Integers

  • Rationals – denominator must be relatively prime to \(p\)

  • FixedMod \(p\)-adics

  • Elements of IntegerModRing(p^k) for k less than or equal to the modulus

The following types should be supported eventually:

  • Finite precision \(p\)-adics

  • Lazy \(p\)-adics

  • Elements of local extensions of THIS \(p\)-adic ring that actually lie in \(\ZZ_p\)

EXAMPLES:

sage: R = Zp(5, 20, 'fixed-mod', 'terse')

Construct from integers:

sage: R(3)
3
sage: R(75)
75
sage: R(0)
0

sage: R(-1)
95367431640624
sage: R(-5)
95367431640620

Construct from rationals:

sage: R(1/2)
47683715820313
sage: R(-7875/874)
9493096742250
sage: R(15/425)
Traceback (most recent call last):
...
ValueError: p divides denominator

Construct from IntegerMod:

sage: R(Integers(125)(3))
3
sage: R(Integers(5)(3))
3
sage: R(Integers(5^30)(3))
3
sage: R(Integers(5^30)(1+5^23))
1
sage: R(Integers(49)(3))
Traceback (most recent call last):
...
TypeError: p does not divide modulus 49

sage: R(Integers(48)(3))
Traceback (most recent call last):
...
TypeError: p does not divide modulus 48

Some other conversions:

sage: R(R(5))
5

Todo

doctests for converting from other types of \(p\)-adic rings

lift()#

Return an integer congruent to self modulo the precision.

Warning

Since fixed modulus elements don’t track their precision, the result may not be correct modulo \(i^{\mathrm{prec_cap}}\) if the element was defined by constructions that lost precision.

EXAMPLES:

sage: R = Zp(7,4,'fixed-mod'); a = R(8); a.lift()
8
sage: type(a.lift())
<class 'sage.rings.integer.Integer'>
multiplicative_order()#

Return the minimum possible multiplicative order of self.

OUTPUT:

an integer – the multiplicative order of this element. This is the minimum multiplicative order of all elements of \(\ZZ_p\) lifting this element to infinite precision.

EXAMPLES:

sage: R = ZpFM(7, 6)
sage: R(1/3)
5 + 4*7 + 4*7^2 + 4*7^3 + 4*7^4 + 4*7^5
sage: R(1/3).multiplicative_order()
+Infinity
sage: R(7).multiplicative_order()
+Infinity
sage: R(1).multiplicative_order()
1
sage: R(-1).multiplicative_order()
2
sage: R.teichmuller(3).multiplicative_order()
6
residue(absprec=1, field=None, check_prec=False)#

Reduce self modulo \(p^\mathrm{absprec}\).

INPUT:

  • absprec – an integer (default: 1)

  • field – boolean (default None). Whether to return an element of GF(p) or Zmod(p).

  • check_prec – boolean (default False). No effect (for compatibility with other types).

OUTPUT:

This element reduced modulo \(p^\mathrm{absprec}\) as an element of \(\ZZ/p^\mathrm{absprec}\ZZ\).

EXAMPLES:

sage: R = Zp(7,4,'fixed-mod')
sage: a = R(8)
sage: a.residue(1)
1

This is different from applying % p^n which returns an element in the same ring:

sage: b = a.residue(2); b
8
sage: b.parent()
Ring of integers modulo 49
sage: c = a % 7^2; c
1 + 7
sage: c.parent()
7-adic Ring of fixed modulus 7^4

See also

_mod_()

class sage.rings.padics.padic_fixed_mod_element.pAdicTemplateElement#

Bases: pAdicGenericElement

A class for common functionality among the \(p\)-adic template classes.

INPUT:

  • parent – a local ring or field

  • x – data defining this element. Various types are supported, including ints, Integers, Rationals, PARI p-adics, integers mod \(p^k\) and other Sage p-adics.

  • absprec – a cap on the absolute precision of this element

  • relprec – a cap on the relative precision of this element

EXAMPLES:

sage: Zp(17)(17^3, 8, 4)
17^3 + O(17^7)
expansion(n=None, lift_mode='simple', start_val=None)#

Return the coefficients in a \(\pi\)-adic expansion. If this is a field element, start at \(\pi^{\mbox{valuation}}\), if a ring element at \(\pi^0\).

For each lift mode, this function returns a list of \(a_i\) so that this element can be expressed as

\[\pi^v \cdot \sum_{i=0}^\infty a_i \pi^i,\]

where \(v\) is the valuation of this element when the parent is a field, and \(v = 0\) otherwise.

Different lift modes affect the choice of \(a_i\). When lift_mode is 'simple', the resulting \(a_i\) will be non-negative: if the residue field is \(\GF{p}\) then they will be integers with \(0 \le a_i < p\); otherwise they will be a list of integers in the same range giving the coefficients of a polynomial in the indeterminant representing the maximal unramified subextension.

Choosing lift_mode as 'smallest' is similar to 'simple', but uses a balanced representation \(-p/2 < a_i \le p/2\).

Finally, setting lift_mode = 'teichmuller' will yield Teichmuller representatives for the \(a_i\): \(a_i^q = a_i\). In this case the \(a_i\) will lie in the ring of integers of the maximal unramified subextension of the parent of this element.

INPUT:

  • n – integer (default None). If given, returns the corresponding entry in the expansion. Can also accept a slice (see slice())

  • lift_mode – 'simple', 'smallest' or 'teichmuller' (default: 'simple')

  • start_val – start at this valuation rather than the default (\(0\) or the valuation of this element).

OUTPUT:

  • If n is None, an iterable giving a \(\pi\)-adic expansion of this element. For base elements the contents will be integers if lift_mode is 'simple' or 'smallest', and elements of self.parent() if lift_mode is 'teichmuller'.

  • If n is an integer, the coefficient of \(\pi^n\) in the \(\pi\)-adic expansion of this element.

Note

Use slice operators to get a particular range.

EXAMPLES:

sage: R = Zp(7,6); a = R(12837162817); a
3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6)
sage: E = a.expansion(); E
7-adic expansion of 3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6)
sage: list(E)
[3, 4, 4, 0, 4, 0]
sage: sum([c * 7^i for i, c in enumerate(E)]) == a
True
sage: E = a.expansion(lift_mode='smallest'); E
7-adic expansion of 3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6) (balanced)
sage: list(E)
[3, -3, -2, 1, -3, 1]
sage: sum([c * 7^i for i, c in enumerate(E)]) == a
True
sage: E = a.expansion(lift_mode='teichmuller'); E
7-adic expansion of 3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6) (teichmuller)
sage: list(E)
[3 + 4*7 + 6*7^2 + 3*7^3 + 2*7^5 + O(7^6),
0,
5 + 2*7 + 3*7^3 + O(7^4),
1 + O(7^3),
3 + 4*7 + O(7^2),
5 + O(7)]
sage: sum(c * 7^i for i, c in enumerate(E))
3 + 4*7 + 4*7^2 + 4*7^4 + O(7^6)

If the element has positive valuation then the list will start with some zeros:

sage: a = R(7^3 * 17)
sage: E = a.expansion(); E
7-adic expansion of 3*7^3 + 2*7^4 + O(7^9)
sage: list(E)
[0, 0, 0, 3, 2, 0, 0, 0, 0]

The expansion of 0 is truncated:

sage: E = R(0, 7).expansion(); E
7-adic expansion of O(7^7)
sage: len(E)
0
sage: list(E)
[]

In fields, on the other hand, the expansion starts at the valuation:

sage: R = Qp(7,4); a = R(6*7+7**2); E = a.expansion(); E
7-adic expansion of 6*7 + 7^2 + O(7^5)
sage: list(E)
[6, 1, 0, 0]
sage: list(a.expansion(lift_mode='smallest'))
[-1, 2, 0, 0]
sage: list(a.expansion(lift_mode='teichmuller'))
[6 + 6*7 + 6*7^2 + 6*7^3 + O(7^4),
2 + 4*7 + 6*7^2 + O(7^3),
3 + 4*7 + O(7^2),
3 + O(7)]

You can ask for a specific entry in the expansion:

sage: a.expansion(1)
6
sage: a.expansion(1, lift_mode='smallest')
-1
sage: a.expansion(2, lift_mode='teichmuller')
2 + 4*7 + 6*7^2 + O(7^3)
lift_to_precision(absprec=None)#

Return another element of the same parent with absolute precision at least absprec, congruent to this \(p\)-adic element modulo the precision of this element.

INPUT:

  • absprec – an integer or None (default: None); the absolute precision of the result. If None, lifts to the maximum precision allowed

Note

If setting absprec that high would violate the precision cap, raises a precision error. Note that the new digits will not necessarily be zero.

EXAMPLES:

sage: R = ZpCA(17)
sage: R(-1,2).lift_to_precision(10)
16 + 16*17 + O(17^10)
sage: R(1,15).lift_to_precision(10)
1 + O(17^15)
sage: R(1,15).lift_to_precision(30)
Traceback (most recent call last):
...
PrecisionError: precision higher than allowed by the precision cap
sage: R(-1,2).lift_to_precision().precision_absolute() == R.precision_cap()
True

sage: R = Zp(5); c = R(17,3); c.lift_to_precision(8)
2 + 3*5 + O(5^8)
sage: c.lift_to_precision().precision_relative() == R.precision_cap()
True

Fixed modulus elements don’t raise errors:

sage: R = ZpFM(5); a = R(5); a.lift_to_precision(7)
5
sage: a.lift_to_precision(10000)
5
residue(absprec=1, field=None, check_prec=True)#

Reduce this element modulo \(p^\mathrm{absprec}\).

INPUT:

  • absprec – 0 or 1.

  • field – boolean (default None). For precision 1, whether to return an element of the residue field or a residue ring. Currently unused.

  • check_prec – boolean (default True). Whether to raise an error if this element has insufficient precision to determine the reduction. Errors are never raised for fixed-mod or floating-point types.

OUTPUT:

This element reduced modulo \(p^\mathrm{absprec}\) as an element of the residue field or the null ring.

EXAMPLES:

sage: # needs sage.libs.ntl
sage: R.<a> = Zq(27, 4)
sage: (3 + 3*a).residue()
0
sage: (a + 1).residue()
a0 + 1
teichmuller_expansion(n=None)#

Returns an iterator over coefficients \(a_0, a_1, \dots, a_n\) such that

  • \(a_i^q = a_i\), where \(q\) is the cardinality of the residue field,

  • this element can be expressed as

\[\pi^v \cdot \sum_{i=0}^\infty a_i \pi^i\]

where \(v\) is the valuation of this element when the parent is a field, and \(v = 0\) otherwise.

  • if \(a_i \ne 0\), the precision of \(a_i\) is \(i\) less than the precision of this element (relative in the case that the parent is a field, absolute otherwise)

Note

The coefficients will lie in the ring of integers of the maximal unramified subextension.

INPUT:

  • n – integer (default None). If given, returns the coefficient of \(\pi^n\) in the expansion.

EXAMPLES:

For fields, the expansion starts at the valuation:

sage: R = Qp(5,5); list(R(70).teichmuller_expansion())
[4 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + O(5^5),
3 + 3*5 + 2*5^2 + 3*5^3 + O(5^4),
2 + 5 + 2*5^2 + O(5^3),
1 + O(5^2),
4 + O(5)]

But if you specify n, you get the coefficient of \(\pi^n\):

sage: R(70).teichmuller_expansion(2)
3 + 3*5 + 2*5^2 + 3*5^3 + O(5^4)
unit_part()#

Returns the unit part of this element.

This is the \(p\)-adic element \(u\) in the same ring so that this element is \(\pi^v u\), where \(\pi\) is a uniformizer and \(v\) is the valuation of this element.

EXAMPLES:

sage: # needs sage.libs.ntl
sage: R.<a> = Zq(125)
sage: (5*a).unit_part()
a + O(5^20)
sage.rings.padics.padic_fixed_mod_element.unpickle_fme_v2(cls, parent, value)#

Unpickles a fixed-mod element.

EXAMPLES:

sage: from sage.rings.padics.padic_fixed_mod_element import pAdicFixedModElement, unpickle_fme_v2
sage: R = ZpFM(5)
sage: a = unpickle_fme_v2(pAdicFixedModElement, R, 17*25); a
2*5^2 + 3*5^3
sage: a.parent() is R
True