A local field L will mean here a finite algebraic extension of some p-adic field Q_p. If n is the degree of the extension L/Q_p, then there exist positive integers e, f, and a subfield T subset L with the properties that n=ef, that T/Q_p of degree f is unramified, and L/T is totally ramified. The integer f is such that the residue class field of L has p^f elements, and is therefore called the residue class degree. The integer e is called the ramification index, and satisfies v(p)=e, where v=v_(pi) is the normalized valuation on L (which has image Z).
Elements of L can be written in the form sum_(k=e)^(Infinity)r_(k)pi^(k) where pi has (normalized) valuation 1 and generates the unique maximal ideal of L, and where the r_k in L are elements of a set of representatives R for the residue classes of O_L/pi O_L isomorphic to F_(p^f).
Local field elements will be stored in general in the form sum_(j=0)^(e - 1)sum_(i=0)^(f - 1)r_(ij)a^ipi^j where a is a generator of the inertia field T=Q_p(a) of degree f over Q_p, and pi is a root of an Eisenstein polynomial of degree e over T. The coefficients r_(ij) are integers satisfying 0 <= r< p^k, where k depends on the precision attached to the field.