NeuchâtelNeuchâtel (UKˌnɜːʃæˈtɛl, US-ʃɑːˈ-,_ˌnjuːʃəˈ-,_ˌnʊʃɑːˈ-, nøʃɑtɛl; Neuenburg) is a town, a municipality, and the capital of the Swiss canton of Neuchâtel on Lake Neuchâtel. Since the fusion in 2021 of the municipalities of Neuchâtel, Corcelles-Cormondrèche, Peseux, and Valangin, the city has approximately 33,000 inhabitants (80,000 in the metropolitan area). The city is sometimes referred to historically by the German name Neuenburg; both the French and German names mean "New Castle".
Galois extensionIn mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension.
PortoPorto or Oporto (ˈpoɾtu) is the second largest city in Portugal after Lisbon, the capital of the Porto District, and one of the Iberian Peninsula's major urban areas. Porto city proper, which is the entire municipality of Porto, is small compared to its metropolitan area, with an estimated population of just 231,800 people in a municipality with only . Porto's metropolitan area has around 1.7 million people (2021) in an area of , making it the second-largest urban area in Portugal.
LausanneLausanne (loʊˈzæn , USalsoloʊˈzɑːn , lɔzan; Losena lɔˈzəna) is the capital and largest city of the Swiss French-speaking canton of Vaud. It is a hilly city situated on the shores of Lake Geneva, about halfway between the Jura Mountains and the Alps, and facing the French town of Évian-les-Bains across the lake. Lausanne is located northeast of Geneva, the nearest major city. The municipality of Lausanne has a population of about 140,000, making it the fourth largest city in Switzerland after Basel, Geneva, and Zurich, with the entire agglomeration area having about 420,000 inhabitants (as of January 2019).
FranceFrance (fʁɑ̃s), officially the French Republic (République française ʁepyblik fʁɑ̃sɛz), is a country located primarily in Western Europe. It also includes overseas regions and territories in the Americas and the Atlantic, Pacific and Indian Oceans, giving it one of the largest discontiguous exclusive economic zones in the world. Its metropolitan area extends from the Rhine to the Atlantic Ocean and from the Mediterranean Sea to the English Channel and the North Sea; overseas territories include French Guiana in South America, Saint Pierre and Miquelon in the North Atlantic, the French West Indies, and many islands in Oceania and the Indian Ocean.
Emmanuel MacronEmmanuel Jean-Michel Frédéric Macron (emanɥɛl makʁɔ̃; born 21 December 1977) is a French politician serving as President of France since 2017. Ex officio, he is also one of the two Co-Princes of Andorra. Earlier, he served as the Minister of Economics, Industry and Digital Affairs under President François Hollande from 2014 to 2016 and Deputy Secretary-General to the President from 2012 to 2014. Born in Amiens, Macron studied philosophy at Paris Nanterre University, later completing a master's degree in public affairs at Sciences Po and graduating from the École nationale d'administration in 2004.
