Sobre o problema inverso de Galois para grupos abelianos finitos

Traballo Fin de Grao en Matemáticas. Curso 2021-2022O Problema Inverso de Galois, no seu enunciado clásico, trata de determinar para que grupos finitos G existe unha extensión de Galois sobre Q que teña grupo de Galois isomorfo a G. Neste traballo tratarase o caso particular no cal o grupo finito sexa ademais abeliano. A exposición do traballo divídese en dúas partes. Na primeira parte probarase que, no caso de grupos abelianos, sempre existe unha extensión que resolve o Problema Inverso de Galois. Na segunda parte demóstrase o Teorema de Kronecker-Weber, o cal establece que toda extensión de Galois de Q con grupo de Galois abeliano é unha extensión ciclotómica. Para os corpos ciclotómicos analizaremos a estrutura dos seus grupos de Galois sobre Q. Ademais, introduciremos a teoría de corpos de números necesaria para a proba do Teorema de Kronecker-Weber, en concreto, presentarase o anel de enteiros dun corpo de números alxébricos, a factorización de ideais primos nunha extensión de corpos de números, así coma, os grupos de descomposición e inercia.The Inverse Galois Problem, in its classic formulation, asks whether given a finite group G there exists a Galois extension over Q whose Galois group is isomorphic to G. This work will address the special case in which the finite group is also abelian. The content of this paper is divided into two parts. In the first part we will prove that, if G is any finite abelian groups, then we can find a Galois extension of Q that solves the Inverse Galois Problem for G. In the second part we will prove the Kronecker-Weber Theorem, which states that if the Galois group of a Galois extension of Q is abelian then the extension is cyclotomic. For cyclotomic fields we study the structure of their Galois groups over Q. Moreover, we will present some concepts of the theory of algebraic number fields that are necessary to prove the Kronecker-Weber Theorem. Specifically, we will introduce the ring of integers of an algebraic number field, the prime decomposition in extensions of number rings and the decomposition and inertia groups

ECAMulticapa: Effectiveness of double-layered compression therapy for healing venous ulcers in primary care: a Study Protocol

Background: Chronic venous insufficiency, in its final stage can cause venous ulcers. Venous ulcers have a prevalence of 0.5 % to 0.8 % in the general population, and increases starting at 60 years of age. This condition often causes increased dependency in affected individuals, as well as a perceived reduced quality of life and family overload. Local Treating chronic venous ulcers has 2 components: topically healing the ulcer and controlling the venous insufficiency. There is evidence that compressive therapy favours the healing process of venous ulcers. The studies we have found suggest that the use of multilayer bandage systems is more effective than the use of bandages with a single component, these are mostly using in Spain. Multilayer compression bandages with 2 layers are equally effective in the healing process of chronic venous ulcers as 4-layer bandages and are better tolerated and preferenced by patients. More studies are needed to specifically compare the 2-layer bandages systems in the settings where these patients are usually treated. Method/design: Randomised, controlled, parallel, multicentre clinical trial, with 12 weeks of follow-up and blind evaluation of the response variable. The objective is to assess the efficacy of multilayer compression bandages (2 layers) compared with crepe bandages, based on the incidence of healed venous ulcers in individuals treated in primary care nursing consultations, at 12 weeks of follow-up. The study will include 216 individuals (108 per branch) with venous ulcers treated in primary care nursing consultations. The primary endpoint is complete healing at 12 weeks of follow-up. The secondary endpoints are the degree of healing (Resvech.2), quality of life (CCVUQ-e), adverse reactions related to the healing process. Prognosis and demographic variables are also recorder. Effectiveness analysis using Kaplan-Meier curves, a log-rank test and a Cox regression analysis. The analysis was performed by intention to treat. Discussion: The study results can contribute to improving the care and quality of life of patients with venous ulcers, decreasing healing times and healthcare expenditure and contributing to the consistent treatment of these lesions. Trial registration: This study has been recorded in the Clinical Trials.gov site with the code NCT02364921. 17 February 2015.This study was funded by PN of I + D + I 2013–2016 and the ISCIII – Subdirección General de Evaluación y Fomento de la Investigación and FEDER funds (PI13/01975). Ministerio de Economia y Competitividad

A pulmonary rehabilitation program reduces hospitalizations in chronic obstructive pulmonary disease patients: A cost-effectiveness study

BACKGROUND: Although pulmonary rehabilitation (PR) is recommended in patients with chronic obstructive pulmonary disease (COPD), there is a scarcity of data demonstrating the cost-effectiveness and effectiveness of PR in reducing exacerbations. METHODS: A quasi-experimental study in 200 patients with COPD was conducted to determine the number of exacerbations 1 year before and after their participation in a PR program. Quality of life was measured using the COPD assessment test and EuroQol-5D. The costs of the program and exacerbations were assessed the year before and after participation in the PR program. The incremental cost-effectiveness ratio (ICER) was estimated in terms of quality-adjusted life years (QALYs). RESULTS: The number of admissions, length of hospital stay, and admissions to the emergency department decreased after participation in the PR program by 48.2%, 46.6%, and 42.5%, respectively (P < 0.001 for all). Results on quality of life tests improved significantly (P < 0.001 for the two tests). The cost of PR per patient and the cost of pre-PR and post-PR exacerbations were €1867.7 and €7895.2 and €4201.9, respectively. The PR resulted in a cost saving of €1826 (total, €365,200) per patient/year, and the gain in QALYs was+0.107. ICER was −€17,056. The total cost was <€20,000/QALY in 78% of patients. Conclusions: PR contributes to reducing the number of exacerbations in patients with COPD, thereby slowing clinical deterioration. In addition, it is cost-effective in terms of QALYs