Ilaria Del Corso

On wild extensions of a p-adic field

Abstract In this paper we consider the problem of classifying the isomorphism classes of extensions of degree p k of a p-adic field K, restricting to the case of extensions without intermediate fields. We establish a correspondence between the isomorphism classes of these extensions and some Kummer extensions of a suitable field F containing K. We then describe such classes in terms of the representations of Gal ( F / K ) . Finally, for k = 2 and for each possible Galois group G, we count the number of isomorphism classes of the extensions whose normal closure has a Galois group isomorphic to G. As a byproduct, we get the total number of isomorphism classes.

On Ore's conjecture and its developments

The p-component of the index of a number field K, indp(K), depends only on the completions of K at the primes over p. More precisely, ind p (K) equals the index of the Qp-algebra K ⊗ Qp. If K is normal, then K ⊗ Q p ≅ L n for some L normal over Qp and some n, and we write Ip(nL) for its index. In this paper we describe an effective procedure to compute Ip(nL) for all n and all normal and tamely ramified extensions L of Qp, hence to determine ind p (K) for all Galois number fields that are tamely ramified at p. Using our procedure, we are able to exhibit a counterexample to a conjecture of Nart (1985) on the behaviour of Ip(nL).

Uniformity over primes of tamely ramified splittings

Abstract:We fix a prime p and let f(X) vary over all monic integer polynomials of fixed degree n. Given any possible shape of a tamely ramified splitting of p in an extension of degree n, we prove that there exists a rational function φ(X)∈ℚ(X) such that the density of the monic integer polynomials f(X) for which the splitting of p has the given shape in ℚ[X]/f(X) is φ(p) (here reducible polynomials can be neglected). As a corollary, we prove that, for p≥n, the density of irreducible monic polynomials of degree n in ℤp[X] is the value at p of a rational function φn(X)∈ℚ(X). All rational functions involved are effectively computable.

Finite groups of units of finite characteristic rings

Fuchs (Abelian groups, Pergamon, Oxford, 1960, Problem 72) asked the following question: which groups can be the group of units of a commutative ring? In the following years, some partial answers have been given to this question in particular cases. The aim of the present paper is to address Fuchs’ question when A is a finite characteristic ring. The result is a pretty good description of the groups which can occur as group of units in this case, equipped with examples showing that there are obstacles to a “short” complete classification. As a by-product, we are able to classify all possible cardinalities of the group of units of a finite characteristic ring, so to answer Ditor’s question (Ditor in Am Math Mon 78(5):522–523, 1971).

Richiami di teoria

Il primo capitolo contiene le note di teoria necessarie quale riferimento per la soluzione dei testi proposti nel libro. In particolare dopo le nozioni fondamentali di insieme, applicazione e relazione, viene richiamato il principio d’induzione. Si passa poi alla divisibilita tra numeri interi e all’aritmetica modulare. La prima struttura algebrica introdotta e quella di gruppo: si studiano i sottogruppi, i prodotti tra gruppi, le classi laterali, i sottogruppi normali e gli omomorfismi tra gruppi. Dopo i gruppi si introducono gli anelli e i loro ideali; particolare attenzione e rivolta alla fattorizzazione dei polinomi, agli anelli di polinomi e ai loro quozienti. Infine si definiscono e si studiano i campi, le loro estensioni e i campi finiti.

Non-invariance of the index in wildly ramified extensions

In this paper, we give an example of three wildly ramified extensions L1, L2, L3 of ℚ2 with the same ramification numbers and isomorphic Galois groups, such that I(nL1) > I(nL2) > I(nL3) for a suitable integer n (where I(nL) denotes the index of the ℚ2-algebra Ln). This example shows that the condition given in [2] for the invariance of the index of tamely ramified extensions is no longer sufficient in the general case.

On the density of the splitting of 2 in cubic fields

SummaryIn this paper we consider all algebraic integers α of degree 3 and all possible splittings of the ideal generated by 2 in cubic fields; we determine the density of the αs such that the splitting of 2 inQ(α) is fixed. We also obtain the density of the integers generating cubic fields of index 2.