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Shafarevich's Theorem On Solvable Galois Groups
In mathematics, Shafarevich's theorem states that any finite solvable group is the Galois group of some finite extension of the rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...s. It was first proved by , though Alexander Schmidt later pointed out a gap in the proof, which was fixed by . References * * Galois theory Solvable groups {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Group
Finite is the opposite of infinite. It may refer to: * Finite number (other) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Groves from the album '' Invisible Empires'' See also * * Nonfinite (other) Nonfinite is the opposite of finite * a nonfinite verb is a verb that is not capable of serving as the main verb in an independent clause * a non-finite clause In linguistics, a non-finite clause is a dependent or embedded clause that represen ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solvable Group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup. Motivation Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable (note this theorem holds only in characteristic 0). This means associated to a polynomial f \in F /math> there is a tower of field extensionsF = F_0 \subseteq F_1 \subseteq F_2 \subseteq \cdots \subseteq F_m=Ksuch that # F_i = F_ alpha_i/math> where \alpha_i^ \in F_, so \alpha_i is a solution to the equation x^ - a where a \in F_ # F_m contains a splitting field for f(x) Example For example, the smallest Galois field extension of \mathbb containing the elemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Group
In mathematics, in the area of abstract 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. For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory. Definition Suppose that E is an extension of the field F (written as E/F and read "''E'' over ''F'' "). An automorphism of E/F is defined to be an automorphism of E that fixes F pointwise. In other words, an automorphism of E/F is an isomorphism \alpha:E\to E such that \alpha(x) = x for each x\in F. The set of all automorphisms of E/F forms a group with the operation of function composition. This group is sometimes denoted by \operatorname(E/F). If E/F is a Galois extension, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Extension
In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — indeed in any area where fields appear prominently. Definition and notation Suppose that ''E''/''F'' is a field extension. Then ''E'' may be considered as a vector space over ''F'' (the field of scalars). The dimension of this vector space is called the degree of the field extension, and it is denoted by :F The degree may be finite or infinite, the field being called a finite extension or infinite extension accordingly. An extension ''E''/''F'' is also sometimes said to be simply finite if it is a finite extension; this should not be confused with the fields themselves being finite fields (fields with finitely many elements). The degree should not be confused with the transcendence degree of a field; for example, the field Q(''X'') o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Schmidt (mathematician)
Alexander Schmidt (born 1965) is a German mathematician at the University of Heidelberg. His research interests include algebraic number theory and algebraic geometry. Life Schmidt attended the Heinrich Heinrich-Hertz-Gymnasium in East Berlin, a special school for mathematics. In 1984 he received the bronze medal at the International Mathematical Olympiad in Prague. In the prologue to his book ''Introduction to Algebraic Number Theory'', he thanks Reinhard Bölling for teaching the reasons why he started at Heinrich-Hertz-Gymnasium. He studied mathematics at the Humboldt University in Berlin and was awarded the diploma in 1991. In 1993, he obtained his PhD at the University of Heidelberg by Kay Wingberg (Positive branched extensions of algebraic number fields). He then was a research assistant and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand. Galois introduced the subject for studying roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is ''solvable by radicals'' if its roots may be expressed by a formula involving only integers, th roots, and the four basic arithmetic operations. This widely generalizes the Abel–Ruffini theorem, which asserts that a general polynomial of degree at least five cannot be solved by radicals. Galois theory has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated (doubling the cub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
