66 p-adic Numbers

returns the family of pure p-adic numbers over the prime p with precision ``digits''.

returns the element of the p-adic number family fam that is used to represent the rational number rat.

p-adic numbers allow the usual operations for fields.

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)

The Valuation is the p-part of the p-adic number.

ShiftedPadicNumber takes a p-adic number padic and an integer shift and returns the p-adic number c, that is padic* p^shift. The shift is just added to the p-part.

66.2 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 neukirch, 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 this subfield.

An extended p-adic field L is given by two polynomials h and g with coeff.-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 an Eisenstein-polynomial (or 1+x). This is not checked by GAP.

Every number out of L is represented as a coeff.-list for the basis {1,x,x²,…,y,xy,x²y,…} 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 one of the polynomials was not irreducible.

create a p-adic number in the p-adic numbers family fam. The first usage returns the p-adic number corresponding to the rational rat.

The second usage takes a pure p-adic numbers family purefam and a list list of length 2 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 usage creates a number in the family extfam of a p-adic extension. The second entry must be a list L of length 2 such that list[2] is the list of coeff. for 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 the coeff.

p-adic numbers allow 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.

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GAP 4 manual
March 2006