query: subclass_of abelian extension
supports: Algebra (from Arabic ""al-jabr"" meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions. In 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.  In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is, in some sense, simpler and better understood. In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them. In 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 . Class field theory provides detailed information about the abelian extensions of number fields , function fields of algebraic curves over finite fields , and local fields . There are two slightly different definitions of the term cyclotomic extension . It can mean either an extension formed by adjoining roots of unity to a field , or a subextension of such an extension . The cyclotomic fields are examples . A cyclotomic extension , under either definition , is always abelian . If a field K contains a primitive n - th root of unity and the n - th root of an element of K is adjoined , the resulting so - called Kummer extension is an abelian extension ( if K has characteristic p we should say that p does n't divide n , since otherwise this can fail even to be a separable extension ) . In general , however , the Galois groups of n - th roots of elements operate both on the n - th roots and on the roots of unity , giving a non-abelian Galois group as semi-direct product . The Kummer theory gives a complete description of the abelian extension case , and the Kronecker -- Weber theorem tells us that if K is the field of rational numbers , an extension is abelian if and only if it is a subfield of a field obtained by adjoining a root of unity . There is an important analogy with the fundamental group in topology , which classifies all covering spaces of a space : abelian covers are classified by its abelianisation which relates directly to the first homology group . In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more finitary operations defined on it that satisfies a list of axioms. Évariste Galois (25 October 1811  31 May 1832) was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a problem standing for 350 years. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra, and the subfield of Galois connections. He died at age 20 from wounds suffered in a duel. In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term "abstract algebra" was coined in the early 20th century to distinguish this area of study from the other parts of algebra. In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object. Mathematics (from Greek  "máthma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. In mathematics, a polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single indeterminate is . An example in three variables is . In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field that contains the base field and satisfies additional properties. For instance, the set Q(2) = {"a" + "b"2 | "a", "b"  Q} is the smallest extension of Q that includes every real solution to the equation "x" = 2.
candidates: age algebra analysis arabic change condition connection definition division engineering field field extension french general geometry greek group integer knowledge learning mathematician mathematics medicine number object problem relationship reunion science sense set solution structure study symmetry group theorem theory via
answer: field extension