Background

The \(p\)-adic field \({\mathbb{Q}}_p\) arises naturally as the completion of \({\mathbb{Q}}\) 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 \(x = p^{v_p(x)} {a \over b}\) but \(p \not \! |\; ab\)). The ring of integers of \({\mathbb{Q}}_p\), denoted \({\mathbb{Z}}_p\), is the set of all elements of non-negative valuation. The ring \({\mathbb{Z}}_p\) has a unique maximal ideal, generated by the prime \(p\); the residue class field \(K\) is the quotient \({\mathbb{Z}}_p / p {\mathbb{Z}}_p\). Any element \(x\) of \({\mathbb{Q}}_p\) can be expressed as a power series in the prime \(p\), so that \(x = \sum_{i=v}^{\infty} a_{i} p^{i}\), 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, \({\mathbb{Q}}_p\) is a local field, with its ring of integers \({\mathbb{Z}}_p\) being a local ring. A uniformizer \(\pi\) of \({\mathbb{Q}}_p\) is the prime \(p\).

More generally, consider an irreducible polynomial over some local field \(L_1\) (such as \({\mathbb{Q}}_p\)). Then the extension given by adjoining a root \(\alpha\) of this polynomial to \(L_1\), \(L_2 = L_1[\alpha]\), is also a local field. Let \(\pi_1\) and \(\pi_2\) be uniformizers of \(L_1\) and \(L_2\), respectively. Then \(\pi_2^e = \pi_1 u\), where \(u\) is a unit of \(L_2\). The number \(e\) is the ramification degree of \(L_2\) over \(L_1\), and divides the degree \(n\) of the extension. If \(e = n\), we say \(L_2\) is totally ramified over \(L_1\); if \(e = 1\), we say \(L_2\) is unramified over \(L_1\). The degree of the residue class field \(K_2\) of \(L_2\) over the residue class field \(K_1\) of \(L_1\) is \(f = {n \over e}\). 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 \(p^n-1\)-th root of unity \(\zeta\) 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 \(\sigma\), which takes \(\zeta\) to \(\zeta^p\). 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 \(f(x) = \sum_{i=0}^{n} a_i x^i\) satisfies \(v_\pi(a_n) = 0\), \(v_\pi(a_i) \ge 1\) for all \(0 < i < n\), and \(v_\pi(a_0) = 1\). 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.