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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 \(\nu_p( p^v \cdot a / b) = v\) 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.

`‣ 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 \(\{ 1, x, x^2, \ldots, y, xy, x^2 y, \ldots \}\) 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 \(\{ 1, \ldots, x^{{f-1}} \cdot y^{{e-1}} \}\) 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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