Re: BIM 360 Field 2017 Portable Torrent
Scat Laboy
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.
The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements.
The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straightedge. Galois theory, devoted to understanding the symmetries of field extensions, provides an elegant proof of the Abel-Ruffini theorem that general quintic equations cannot be solved in radicals.
Fields serve as foundational notions in several mathematical domains. This includes different branches of mathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Number fields, the siblings of the field of rational numbers, are studied in depth in number theory. Function fields can help describe properties of geometric objects.
An equivalent, and more succinct, definition is: a field has two commutative operations, called addition and multiplication; it is a group under addition with 0 as the additive identity; the nonzero elements are a group under multiplication with 1 as the multiplicative identity; and multiplication distributes over addition.
In antiquity, several geometric problems concerned the (in)feasibility of constructing certain numbers with compass and straightedge. For example, it was unknown to the Greeks that it is, in general, impossible to trisect a given angle in this way. These problems can be settled using the field of constructible numbers.[7] Real constructible numbers are, by definition, lengths of line segments that can be constructed from the points 0 and 1 in finitely many steps using only compass and straightedge. These numbers, endowed with the field operations of real numbers, restricted to the constructible numbers, form a field, which properly includes the field Q of rational numbers. The illustration shows the construction of square roots of constructible numbers, not necessarily contained within Q. Using the labeling in the illustration, construct the segments AB, BD, and a semicircle over AD (center at the midpoint C), which intersects the perpendicular line through B in a point F, at a distance of exactly h = p \displaystyle h=\sqrt p from B when BD has length one.
Not all real numbers are constructible. It can be shown that 2 3 \displaystyle \sqrt[3]2 is not a constructible number, which implies that it is impossible to construct with compass and straightedge the length of the side of a cube with volume 2, another problem posed by the ancient Greeks.
is compatible with the addition in F (and also with the multiplication), and is therefore a field homomorphism.[12] The existence of this homomorphism makes fields in characteristic p quite different from fields of characteristic 0.
Historically, three algebraic disciplines led to the concept of a field: the question of solving polynomial equations, algebraic number theory, and algebraic geometry.[17] A first step towards the notion of a field was made in 1770 by Joseph-Louis Lagrange, who observed that permuting the zeros x1, x2, x3 of a cubic polynomial in the expression
(with ω being a third root of unity) only yields two values. This way, Lagrange conceptually explained the classical solution method of Scipione del Ferro and Franois Vite, which proceeds by reducing a cubic equation for an unknown x to a quadratic equation for x3.[18] Together with a similar observation for equations of degree 4, Lagrange thus linked what eventually became the concept of fields and the concept of groups.[19] Vandermonde, also in 1770, and to a fuller extent, Carl Friedrich Gauss, in his Disquisitiones Arithmeticae (1801), studied the equation
for a prime p and, again using modern language, the resulting cyclic Galois group. Gauss deduced that a regular p-gon can be constructed if p = 22k + 1. Building on Lagrange's work, Paolo Ruffini claimed (1799) that quintic equations (polynomial equations of degree 5) cannot be solved algebraically; however, his arguments were flawed. These gaps were filled by Niels Henrik Abel in 1824.[20] variste Galois, in 1832, devised necessary and sufficient criteria for a polynomial equation to be algebraically solvable, thus establishing in effect what is known as Galois theory today. Both Abel and Galois worked with what is today called an algebraic number field, but conceived neither an explicit notion of a field, nor of a group.
In 1871 Richard Dedekind introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, the German word Krper, which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by Moore (1893).[21]
By a field we will mean every infinite system of real or complex numbers so closed in itself and perfect that addition, subtraction, multiplication, and division of any two of these numbers again yields a number of the system.
The first clear definition of an abstract field is due to Weber (1893).[24] In particular, Heinrich Martin Weber's notion included the field Fp. Giuseppe Veronese (1891) studied the field of formal power series, which led Hensel (1904) to introduce the field of p-adic numbers. Steinitz (1910) synthesized the knowledge of abstract field theory accumulated so far. He axiomatically studied the properties of fields and defined many important field-theoretic concepts. The majority of the theorems mentioned in the sections Galois theory, Constructing fields and Elementary notions can be found in Steinitz's work. Artin & Schreier (1927) linked the notion of orderings in a field, and thus the area of analysis, to purely algebraic properties.[25] Emil Artin redeveloped Galois theory from 1928 through 1942, eliminating the dependency on the primitive element theorem.
In the hierarchy of algebraic structures fields can be characterized as the commutative rings R in which every nonzero element is a unit (which means every element is invertible). Similarly, fields are the commutative rings with precisely two distinct ideals, (0) and R. Fields are also precisely the commutative rings in which (0) is the only prime ideal.
A field extension in which every element of F is algebraic over E is called an algebraic extension. Any finite extension is necessarily algebraic, as can be deduced from the above multiplicativity formula.[31]
whose value is greater than that element, that is, there are no infinite elements. Equivalently, the field contains no infinitesimals (elements smaller than all rational numbers); or, yet equivalent, the field is isomorphic to a subfield of R.
An ordered field is Dedekind-complete if all upper bounds, lower bounds (see Dedekind cut) and limits, which should exist, do exist. More formally, each bounded subset of F is required to have a least upper bound. Any complete field is necessarily Archimedean,[38] since in any non-Archimedean field there is neither a greatest infinitesimal nor a least positive rational, whence the sequence 1/2, 1/3, 1/4, ..., every element of which is greater than every infinitesimal, has no limit.
Since every proper subfield of the reals also contains such gaps, R is the unique complete ordered field, up to isomorphism.[39] Several foundational results in calculus follow directly from this characterization of the reals.
The hyperreals R* form an ordered field that is not Archimedean. It is an extension of the reals obtained by including infinite and infinitesimal numbers. These are larger, respectively smaller than any real number. The hyperreals form the foundational basis of non-standard analysis.
Differential fields are fields equipped with a derivation, i.e., allow to take derivatives of elements in the field.[44] For example, the field R(X), together with the standard derivative of polynomials forms a differential field. These fields are central to differential Galois theory, a variant of Galois theory dealing with linear differential equations.
Galois theory studies algebraic extensions of a field by studying the symmetry in the arithmetic operations of addition and multiplication. An important notion in this area is that of finite Galois extensions F / E, which are, by definition, those that are separable and normal. The primitive element theorem shows that finite separable extensions are necessarily simple, i.e., of the form
The set of such formulas for all n expresses that E is algebraically closed.The Lefschetz principle states that C is elementarily equivalent to any algebraically closed field F of characteristic zero. Moreover, any fixed statement φ holds in C if and only if it holds in any algebraically closed field of sufficiently high characteristic.[50]
since it behaves in several ways as a limit of the fields Fi: Łoś's theorem states that any first order statement that holds for all but finitely many Fi, also holds for the ultraproduct. Applied to the above sentence φ, this shows that there is an isomorphism[e]
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