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In this chapter p is always a (fixed) prime integer.

The p-adic numbers Q_p are the completion of the rational numbers with respect to the valuation ν_p( p^v ⋅ a / b) = v if p divides neither a nor b. They form a field of characteristic 0 which nevertheless shows some behaviour of the finite field with p elements.

A p-adic numbers can be represented by a "p-adic expansion" which is similar to the decimal expansion used for the reals (but written from left to right). So for example if p = 2, the numbers 1, 2, 3, 4, 1/2, and 4/5 are represented as 1(2), 0.1(2), 1.1(2), 0.01(2), 10(2), and the infinite periodic expansion 0.010110011001100...(2). p-adic numbers can be approximated by ignoring higher powers of p, so for example with only 2 digits accuracy 4/5 would be approximated as 0.01(2). This is different from the decimal approximation of real numbers in that p-adic approximation is a ring homomorphism on the subrings of p-adic numbers whose valuation is bounded from below so that rounding errors do not increase with repeated calculations.

In **GAP**, p-adic numbers are always represented by such approximations. A family of approximated p-adic numbers consists of p-adic numbers with a fixed prime p and a certain precision, and arithmetic with these numbers is done with this precision.

Pure p-adic numbers are the p-adic numbers described so far.

`‣ PurePadicNumberFamily` ( p, precision ) | ( function ) |

returns the family of pure p-adic numbers over the prime `p` with `precision` "digits". That is to say, the approximate value will differ from the correct value by a multiple of p^digits.

`‣ PadicNumber` ( fam, rat ) | ( operation ) |

returns the element of the p-adic number family `fam` that approximates the rational number `rat`.

p-adic numbers allow the usual operations for fields.

gap> fam:=PurePadicNumberFamily(2,20);; gap> a:=PadicNumber(fam,4/5); 0.010110011001100110011(2) gap> fam:=PurePadicNumberFamily(2,3);; gap> a:=PadicNumber(fam,4/5); 0.0101(2) gap> 3*a; 0.0111(2) gap> a/2; 0.101(2) gap> a*10; 0.001(2)

See `PadicNumber`

(68.2-2) for other methods for `PadicNumber`

.

`‣ Valuation` ( obj ) | ( operation ) |

The valuation is the p-part of the p-adic number. See also `PValuation`

(15.7-1).

`‣ ShiftedPadicNumber` ( padic, int ) | ( operation ) |

`ShiftedPadicNumber`

takes a p-adic number `padic` and an integer `shift` and returns the p-adic number c, that is `padic` `*`

p`^`

`shift`.

`‣ IsPurePadicNumber` ( obj ) | ( category ) |

The category of pure p-adic numbers.

`‣ IsPurePadicNumberFamily` ( fam ) | ( category ) |

The family of pure p-adic numbers.

The usual Kronecker construction with an irreducible polynomial can be used to construct extensions of the p-adic numbers. Let L be such an extension. Then there is a subfield K < L such that K is an unramified extension of the p-adic numbers and L/K is purely ramified.

(For an explanation of "ramification" see for example [Neu92, Section II.7], or another book on algebraic number theory. Essentially, an extension L of the p-adic numbers generated by a rational polynomial f is unramified if f remains squarefree modulo p and is completely ramified if modulo p the polynomial f is a power of a linear factor while remaining irreducible over the p-adic numbers.)

The representation of extensions of p-adic numbers in **GAP** uses the subfield K.

`‣ PadicExtensionNumberFamily` ( p, precision, unram, ram ) | ( function ) |

An extended p-adic field L is given by two polynomials h and g with coefficient lists `unram` (for the unramified part) and `ram` (for the ramified part). Then L is isomorphic to Q_p[x,y]/(h(x),g(y)).

This function takes the prime number `p` and the two coefficient lists `unram` and `ram` for the two polynomials. The polynomial given by the coefficients in `unram` must be a cyclotomic polynomial and the polynomial given by `ram` must be either an Eisenstein polynomial or 1+x. *This is not checked by GAP.*

Every number in L is represented as a coefficient list w. r. t. the basis { 1, x, x^2, ..., y, xy, x^2 y, ... } of L. The integer `precision` is the number of "digits" that all the coefficients have.

*A general comment:*

The polynomials with which `PadicExtensionNumberFamily`

is called define an extension of Q_p. It must be ensured that both polynomials are really irreducible over Q_p! For example x^2+x+1 is *not* irreducible over Q_p. Therefore the "extension" `PadicExtensionNumberFamily(3, 4, [1,1,1], [1,1])`

contains non-invertible "pseudo-p-adic numbers". Conversely, if an "extension" contains noninvertible elements then one of the defining polynomials was not irreducible.

`‣ PadicNumber` ( fam, rat ) | ( operation ) |

`‣ PadicNumber` ( purefam, list ) | ( operation ) |

`‣ PadicNumber` ( extfam, list ) | ( operation ) |

(see also `PadicNumber`

(68.1-2)).

`PadicNumber`

creates a p-adic number in the p-adic numbers family `fam`. The first form returns the p-adic number corresponding to the rational `rat`.

The second form takes a pure p-adic numbers family `purefam` and a list `list` of length two, and returns the number p`^`

`list``[1] * `

`list``[2]`

. It must be guaranteed that no entry of `list``[2]`

is divisible by the prime p. (Otherwise precision will get lost.)

The third form creates a number in the family `extfam` of a p-adic extension. The second argument must be a list `list` of length two such that `list``[2]`

is the list of coefficients w.r.t. the basis { 1, ..., x^{f-1} ⋅ y^{e-1} } of the extended p-adic field and `list``[1]`

is a common p-part of all these coefficients.

p-adic numbers admit the usual field operations.

gap> efam:=PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1]);; gap> PadicNumber(efam,7/9); padic(120(3),0(3))

*A word of warning:*

Depending on the actual representation of quotients, precision may seem to "vanish". For example in `PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1])`

the number `(1.2000, 0.1210)(3)`

can be represented as `[ 0, [ 1.2000, 0.1210 ] ]`

or as `[ -1, [ 12.000, 1.2100 ] ]`

(here the coefficients have to be multiplied by p^{-1}).

So there may be a number `(1.2, 2.2)(3)`

which seems to have only two digits of precision instead of the declared 5. But internally the number is stored as `[ -3, [ 0.0012, 0.0022 ] ]`

and so has in fact maximum precision.

`‣ IsPadicExtensionNumber` ( obj ) | ( category ) |

The category of elements of the extended p-adic field.

gap> efam:=PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1]);; gap> IsPadicExtensionNumber(PadicNumber(efam,7/9)); true

`‣ IsPadicExtensionNumberFamily` ( fam ) | ( category ) |

Family of elements of the extended p-adic field.

gap> efam:=PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1]);; gap> IsPadicExtensionNumberFamily(efam); true

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