|
Artin–Schreier Theory
In mathematics, Artin–Schreier theory is a branch of Galois theory, specifically a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic ''p''. introduced Artin–Schreier theory for extensions of prime degree ''p'', and generalized it to extensions of prime power degree ''p''''n''. If ''K'' is a field of characteristic ''p'', a prime number, any polynomial of the form :X^p - X - \alpha,\, for \alpha in ''K'', is called an ''Artin–Schreier polynomial''. When \alpha\neq \beta^p-\beta for all \beta \in K, this polynomial is irreducible in ''K'' 'X'' and its splitting field over ''K'' is a cyclic extension of ''K'' of degree ''p''. This follows since for any root ''β'', the numbers ''β'' + ''i'', for 1\le i\le p, form all the roots—by Fermat's little theorem—so the splitting field is K(\beta) . Conversely, any Galois extension of ''K'' of degree ''p'' equal to the characteristic of ''K'' is the splitting field ... [...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] |
Fermat's Little Theorem
Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as : a^p \equiv a \pmod p. For example, if = 2 and = 7, then 27 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7. If is not divisible by , that is if is coprime to , Fermat's little theorem is equivalent to the statement that is an integer multiple of , or in symbols: : a^ \equiv 1 \pmod p. For example, if = 2 and = 7, then 26 = 64, and 64 − 1 = 63 = 7 × 9 is thus a multiple of 7. Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's Last Theorem.. History Pierre de Fermat first stated the theorem in a letter dated October ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Witt Vector
In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors W(\mathbb_p) over the finite field of order p is the ring of p-adic integers. They have a highly non-intuitive structure upon first glance because their additive and multiplicative structure depends on an infinite set of recursive formulas which do not behave like addition and multiplication formulas for standard p-adic integers. The main idea behind Witt vectors is instead of using the standard p-adic expansiona = a_0+a_1p+a_2p^2 + \cdotsto represent an element in \mathbb_p, we can instead consider an expansion using the Teichmüller character\omega: \mathbb_p^* \to \mathbb_p^*which sends each element in the solution set of x^-1 in \mathbb_p to an element in the solution set of x^-1 in \mathbb_p. That is, we expand out elements in \mathbb_p in terms of roots of unity instead of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Purely Inseparable Extension
In algebra, a purely inseparable extension of fields is an extension ''k'' ⊆ ''K'' of fields of characteristic ''p'' > 0 such that every element of ''K'' is a root of an equation of the form ''x''''q'' = ''a'', with ''q'' a power of ''p'' and ''a'' in ''k''. Purely inseparable extensions are sometimes called radicial extensions, which should not be confused with the similar-sounding but more general notion of radical extensions. Purely inseparable extensions An algebraic extension E\supseteq F is a ''purely inseparable extension'' if and only if for every \alpha\in E\setminus F, the minimal polynomial of \alpha over ''F'' is ''not'' a separable polynomial.Isaacs, p. 298 If ''F'' is any field, the trivial extension F\supseteq F is purely inseparable; for the field ''F'' to possess a ''non-trivial'' purely inseparable extension, it must be imperfect as outlined in the above section. Several equivalent and more concrete definitions for the notion of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isogeny
In mathematics, in particular, in algebraic geometry, an isogeny is a morphism of algebraic groups (also known as group varieties) that is surjective and has a finite kernel. If the groups are abelian varieties, then any morphism of the underlying algebraic varieties which is surjective with finite fibres is automatically an isogeny, provided that . Such an isogeny then provides a group homomorphism between the groups of -valued points of and , for any field over which is defined. The terms "isogeny" and "isogenous" come from the Greek word ισογενη-ς, meaning "equal in kind or nature". The term "isogeny" was introduced by Weil; before this, the term "isomorphism" was somewhat confusingly used for what is now called an isogeny. Case of abelian varieties For abelian varieties, such as elliptic curves, this notion can also be formulated as follows: Let ''E''1 and ''E''2 be abelian varieties of the same dimension over a field ''k''. An isogeny between ''E''1 and ''E''2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Varieties
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined ''over'' that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization techniques lead naturally from abe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solvability By Radicals
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] |
Galois Cohomology
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a natural way on some abelian groups, for example those constructed directly from ''L'', but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor. History The current theory of Galois cohomology came together around 1950, when it was realised that the Galois cohomology of ideal class groups in algebraic number theory was one way to formulate class field theory, at the time it was in the process of ridding itself of connections to L-functions. Galois cohomology makes no assumption that Galois groups are abelian groups, so this was a non-abelian theory. It was formulated abstractly as a theory of class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Theorem 90
In abstract algebra, Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if ''L''/''K'' is an extension of fields with cyclic Galois group ''G'' = Gal(''L''/''K'') generated by an element \sigma, and if a is an element of ''L'' of relative norm 1, that isN(a):=a\, \sigma(a)\, \sigma^2(a)\cdots \sigma^(a)=1,then there exists b in ''L'' such thata=b/\sigma(b).The theorem takes its name from the fact that it is the 90th theorem in David Hilbert's Zahlbericht , although it is originally due to . Often a more general theorem due to is given the name, stating that if ''L''/''K'' is a finite Galois extension of fields with arbitrary Galois group ''G'' = Gal(''L''/''K''), then the first cohomology group of ''G'', with coefficients in the multiplicative group of ''L'', is trivial: :H^1(G,L^\times)=\. Examples Let L/K be the quad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Extension
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 ro ... [...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] |
Splitting Field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a polynomial ''p''(''X'') over a field ''K'' is a field extension ''L'' of ''K'' over which ''p'' factors into linear factors :p(X) = c\prod_^ (X - a_i) where c\in K and for each i we have X - a_i \in L /math> with ''ai'' not necessarily distinct and such that the roots ''ai'' generate ''L'' over ''K''. The extension ''L'' is then an extension of minimal degree over ''K'' in which ''p'' splits. It can be shown that such splitting fields exist and are unique up to isomorphism. The amount of freedom in that isomorphism is known as the Galois group of ''p'' (if we assume it is separable). Properties An extension ''L'' which is a splitting field for a set of polynomials ''p''(''X'') over ''K'' is called a normal extension of ''K''. Given an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
