Subsections

• Generic Element Constructions
• Element Constructors and Selectors
• Sequence Conversions

The usual functions for zero and unity in a ring work for p-adic fields and rings as well. Note that Zero creates the only infinity precision element in the fixed precision rings.

Zero(R) : RngPad -> RngPadElt

One(R) : RngPad -> RngPadElt

Zero(R) : FldPad -> FldPadElt

One(R) : FldPad -> FldPadElt

Random(R) : RngPad -> RngPadElt

Representative(R) : RngPad -> RngPad

R . 1 : RngLoc -> RngLocElt

Returns a `uniformizing parameter' for R (of valuation 1), which will be the element p with infinite precision in Z_p or Q_p, it will be p + O(p^(r)) in Z_p/p^r Z_p, and p + O(p^(r + 1)) in Q_p^((r)).

R ! a : FldLoc, RngElt -> FldLocElt

elt<R | a> : FldLoc, FldRat -> FldLocElt

R ! a : FldLoc, FldRat -> FldLocElt

elt<R | a> : RngLoc, RngElt -> RngLocElt

R ! a : RngLoc, RngElt -> RngLocElt

elt<R | a> : RngLoc, FldRat -> RngLocElt

R ! a : RngLoc, FldRat -> RngLocElt

Given an element a, coerce it into the local field or ring R. The resulting element will have either infinite absolute precision (if R is Z_p or Q_p) or the fixed precision associated to the field or ring. The element a is allowed to be one of the following

• an integer;
• a rational number (of valuation 1 if R is Z_p);
• a sequence of integers (or integral rational numbers) in the range from 0 to p - 1;
• a sequence of elements of Z/pZ;
• a sequence of elements of GF(p).

Any sequence of elements will be taken as coefficients in the p-adic expansion.

elt<R | v, a> : FldLoc, RngIntElt, RngElt -> FldLocElt

elt<R | a, r> : FldLoc, RngIntElt, RngElt -> FldLocElt

elt<R | v, a, r> : FldLoc, RngIntElt, RngElt -> FldLocElt

Create the element p^va + O(p^r) in R determined by the sequence a (consisting of elements of Z, Z/pZ or GF(p)) as well as one or both of the integers v (the valuation) and r (the precision). If v is omitted, it is taken to be zero, if r is omitted it is taken to be v + l, where l is the length of the sequence a.

O(x^n) : RngElt -> RngIntElt

Create the series 0 + O(x^n) where x is the `generator' of a local ring R (and where n must be positive if R is Z_p).

Seqelt(s, F) : [ FldLocElt ] -> FldLocElt

Given a sequence s=[s_1, ..., s_l] of integers in the range 0, ..., p - 1, or of integers modulo p, or of elements of the prime field GF(p), create the element s_1 + s_1p + ... + s_(l)p^(l - 1) + O(p^l) in R, where R must be a p-adic ring.

ElementToSequence(a) : FldLocElt -> [ RngResElt ]

Eltseq(a) : FldLocElt -> [ FldLocElt ]

Given an element a=a_0 + a_1p + ... + a_(s - 1)p^(s - 1) + O(p^s) of some p-adic ring, return the sequence s of its coefficients as elements of Z/pZ.