Background

The p-adic field Q_p arises naturally as the completion of Q with respect to an absolute value function |x|_p = p ^{- v_p(x)}, where v_p(x) is the p-adic valuation of x (that is, a power of p such that x = p^v_p(x) (a /b) but p not | ab). The ring of integers of Q_p, denoted Z_p, is the set of all elements of non-negative valuation. The ring Z_p has a unique maximal ideal, generated by the prime p; the residue class field K is the quotient Z_p / p Z_p. Any element x of Q_p can be expressed as a power series in the prime p, so that x = ∑_i=v^∞ a_i pⁱ, where v is the valuation of x, a_v is non-zero, and each a_i is a lift of an element from the residue class field. In more general terms, Q_p is a local field, with its ring of integers Z_p being a local ring. A uniformizer π of Q_p is the prime p.

More generally, consider an irreducible polynomial over some local field L₁ (such as Q_p). Then the extension given by adjoining a root α of this polynomial to L₁, L₂ = L₁[α], is also a local field. Let π₁ and π₂ be uniformizers of L₁ and L₂, respectively. Then π₂^e = π₁ u, where u is a unit of L₂. The number e is the ramification degree of L₂ over L₁, and divides the degree n of the extension. If e = n, we say L₂ is totally ramified over L₁; if e = 1, we say L₂ is unramified over L₁. The degree of the residue class field K₂ of L₂ over the residue class field K₁ of L₁ is f = (n /e). Finite extensions in Magma must be either unramified or totally ramified; Magma also allows towers of extensions to be built.

It is well known that up to isomorphism there is only one degree n unramified extension of L, which can be obtained by adjoining a pⁿ - 1-th root of unity ζ to L. This extension is Galois, with Galois group isomorphic to the cyclic group of order n. The Galois group is generated by the Frobenius automorphism σ, which takes ζ to ζ^p. For some applications, it is necessary to have a fast Frobenius action, so Magma supports such a representation. However, the defining polynomial in this representation is particularly dense and can be expensive to construct, hence it is ill-suited for general applications. An unramified extension can only be defined by inertial polynomials, which are polynomials that keep their degree when mapped over the residue class field and are irreducible over the residue class field. Magma allows an unramified extension to be defined by any inertial polynomial.

All totally ramified extensions contain a root of an Eisenstein polynomial of L. An Eisenstein polynomial f(x) = ∑_i=0ⁿ a_i xⁱ satisfies v_π(a_n) = 0, v_π(a_i) ≥1 for all 0 < i < n, and v_π(a₀) = 1. Magma allows a totally ramified extension to be defined by any Eisenstein polynomial; note that this does not allow arbitrary representations of totally ramified extensions to be constructed. [Next][Prev] [Right] [Left] [Up] [Index] [Root]