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68 p-adic Numbers (preliminary)

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.

68.1 Pure p-adic Numbers

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

68.1-1 PurePadicNumberFamily

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 $p^{digits}$.

68.1-2 PadicNumber

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.

68.1-3 Valuation

The valuation is the $p$-part of the $p$-adic number. See also PValuation (15.7-1).

68.1-4 ShiftedPadicNumber

ShiftedPadicNumber takes a $p$-adic number padic and an integer shift and returns the $p$-adic number $c$, that is padic * $p$^shift.

68.1-5 IsPurePadicNumber

The category of pure $p$-adic numbers.

68.1-6 IsPurePadicNumberFamily

The family of pure $p$-adic numbers.

68.2 Extensions of the 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 $K$.

68.2-1 PadicExtensionNumberFamily

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.

68.2-2 PadicNumber

(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.

68.2-3 IsPadicExtensionNumber

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

68.2-4 IsPadicExtensionNumberFamily

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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