Abelian extensionIn abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable, i.e., if the group can be decomposed into a series of normal extensions of an abelian group. Every finite extension of a finite field is a cyclic extension.
Field extensionIn mathematics, particularly in algebra, a field extension is a pair of fields such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry.
Separable extensionIn field theory, a branch of algebra, an algebraic field extension is called a separable extension if for every , the minimal polynomial of over F is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated roots in any extension field). There is also a more general definition that applies when E is not necessarily algebraic over F. An extension that is not separable is said to be inseparable.
Group extensionIn mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If and are two groups, then is an extension of by if there is a short exact sequence If is an extension of by , then is a group, is a normal subgroup of and the quotient group is isomorphic to the group . Group extensions arise in the context of the extension problem, where the groups and are known and the properties of are to be determined.
Normal extensionIn abstract algebra, a normal extension is an algebraic field extension L/K for which every irreducible polynomial over K which has a root in L, splits into linear factors in L. These are one of the conditions for algebraic extensions to be a Galois extension. Bourbaki calls such an extension a quasi-Galois extension. Let be an algebraic extension (i.e. L is an algebraic extension of K), such that (i.e. L is contained in an algebraic closure of K).
WalloniaWallonia (wɒˈloʊniə; Wallonie walɔni), officially the Walloon Region (Région wallonne), is one of the three regions of Belgium—along with Flanders and Brussels. Covering the southern portion of the country, Wallonia is primarily French-speaking. It accounts for 55% of Belgium's territory, but only a third of its population. The Walloon Region and the French Community of Belgium, which is the political entity responsible for matters related mainly to culture and education, are independent concepts, because the French Community of Belgium encompasses both Wallonia and the bilingual Brussels-Capital Region.
Democratic Republic of the CongoThe Democratic Republic of the Congo (DRC), also known as Congo-Kinshasa and formerly known as Zaire, is a country in Central Africa. By land area, the DRC is the second-largest country in Africa and the 11th-largest in the world. With a population of around 112 million, the Democratic Republic of the Congo is the most populous officially Francophone country in the world. The national capital and largest city is Kinshasa, which is also the economic center.
KinshasaKinshasa (kɪnˈʃɑːsə; kinʃasa; Kinsásá), formerly Léopoldville, is the capital and largest city of the Democratic Republic of the Congo. Once a site of fishing and trading villages situated along the Congo River, Kinshasa is now one of the world's fastest growing megacities. The city of Kinshasa is also one of the DRC's 26 provinces. Because the administrative boundaries of the city-province cover a vast area, over 90 per cent of the city-province's land is rural in nature, and the urban area occupies a small but expanding section on the western side.
BrusselsBrussels (Bruxelles bʁysɛl or bʁyksɛl; Brussel ˈbrʏsəl), officially the Brussels-Capital Region (Région de Bruxelles-Capitale; Brussels Hoofdstedelijk Gewest), is a region of Belgium comprising 19 municipalities, including the City of Brussels, which is the capital of Belgium. The Brussels-Capital Region is located in the central portion of the country and is a part of both the French Community of Belgium and the Flemish Community, but is separate from the Flemish Region (within which it forms an enclave) and the Walloon Region.
