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André Plane
In mathematics, André planes are a class of finite translation planes found by André. The Desarguesian plane and the Hall planes are examples of André planes; the two-dimensional regular Near-field (mathematics), nearfield planes are also André planes. Construction Let F = GF(q) be a finite Field (mathematics), field, and let K = GF(q^n) be a degree n field extension, extension field of F. Let \Gamma be the group of field automorphisms of K over F, and let \beta be an arbitrary mapping from F to \Gamma such that \beta(1)=1. Finally, let N be the Field norm, norm function from K to F. Define a quasifield Q with the same elements and addition as K, but with multiplication defined via a \circ b = a^ \cdot b, where \cdot denotes the normal field multiplication in K. Using this quasifield to Translation plane#Algebraic Construction with Coordinates, construct a plane yields an André plane. Properties # André planes exist for all proper prime powers p^n with p prime and n a positiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation Plane
In mathematics, a translation plane is a projective plane which admits a certain group of symmetries (described below). Along with the Hughes planes and the Figueroa planes, translation planes are among the most well-studied of the known non-Desarguesian planes, and the vast majority of known non-Desarguesian planes are either translation planes, or can be obtained from a translation plane via successive iterations of Duality (projective geometry), dualization and/or Hall plane#Derivation, derivation. In a projective plane, let represent a point, and represent a line. A ''Homography#Central collineations, central collineation'' with ''center'' and ''axis'' is a collineation fixing every point on and every line through . It is called an ''elation'' if is on , otherwise it is called a ''homology''. The central collineations with center and axis form a group. A line in a projective plane is a ''translation line'' if the group of all elations with axis Group action (mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Zeitschrift
''Mathematische Zeitschrift'' ( German for ''Mathematical Journal'') is a mathematical journal for pure and applied mathematics published by Springer Verlag. History The journal was founded in 1917, with its first issue appearing in 1918. It was initially edited by Leon Lichtenstein together with Konrad Knopp, Erhard Schmidt, and Issai Schur. Because Lichtenstein was Jewish, he was forced to step down as editor in 1933 under the Nazi rule of Germany; he fled to Poland and died soon after. The editorship was offered to Helmut Hasse Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and ..., but he refused, Translated by Bärbel Deninger from the 1982 German original. and Konrad Knopp took it over. Other past editors include Erich Kamke, Friedrich Karl Schmidt, Rolf Nevanlinna, Hel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Desarguesian Plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus ''any'' two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by , RP2, or P2(R), among other notations. There are many other projective planes, both infinite, such as the complex projective plane, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hall Plane
In mathematics, a Hall plane is a non-Desarguesian projective plane constructed by Marshall Hall Jr. (1943). There are examples of order ''p''2''n'' for every prime ''p'' and every positive integer ''n'' provided . Algebraic construction via Hall systems The original construction of Hall planes was based on the Hall quasifield (also called a ''Hall system''), ''H'' of order ''p''2''n'' for ''p'' a prime. The creation of the plane from the quasifield follows the standard construction (see quasifield for details). To build a Hall quasifield, start with a Galois field, for ''p'' a prime and a quadratic irreducible polynomial over ''F''. Extend , a two-dimensional vector space over ''F'', to a quasifield by defining a multiplication on the vectors by when and otherwise. Writing the elements of ''H'' in terms of a basis , that is, identifying with as ''x'' and ''y'' vary over ''F'', we can identify the elements of ''F'' as the ordered pairs , i.e. . The properties of the defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Near-field (mathematics)
In mathematics, a near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws. Alternatively, a near-field is a near-ring in which there is a multiplicative identity and every non-zero element has a multiplicative inverse. Definition A near-field is a set Q together with two binary operations, + (addition) and \cdot (multiplication), satisfying the following axioms for all a, b, c in Q . :A1: (Q, +) is an abelian group. :A2: (a \cdot b) \cdot c = a \cdot (b \cdot c) (The associative law for multiplication). :A3: (a + b) \cdot c = a \cdot c + b \cdot c (The right distributive law). :A4: Q contains a non-zero element 1 such that 1 \cdot a = a \cdot 1 = a (Multiplicative identity). :A5: For every non-zero element d in Q there exists an element d^ such that d \cdot d^ = 1 = d^ \cdot d (Multiplicative inverse). Notes on the definition # The above is, strictly speaking, a definition of a ''right'' near-field. By rep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, 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 field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''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 element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Extension
In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, 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. Subfield A subfield K of a field L is a subset K\subseteq L that is a field with respect to the field operations inherited from L. Equivalently, a subfield is a subset that contains the multiplicative identity 1, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of K. As , the latter definition implies K and L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Automorphism
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. Definition In an algebraic structure such as a group, a ring, or vector space, an ''automorphism'' is simply a bijective homomorphism of an object into itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator.) More generally, for an object in some category, an automorphism is a morphism of the object to itself that has an inverse morphism; that is, a morphism f: X\to X is an automorphism if there is a morphism g: X\to X such that g\circ f= f\circ g = \operatorname _X, where \operatorname _X is the identity morph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Norm
In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield. Formal definition Let ''K'' be a field and ''L'' a finite extension (and hence an algebraic extension) of ''K''. The field ''L'' is then a finite-dimensional vector space over ''K''. Multiplication by ''α'', an element of ''L'', :m_\alpha\colon L\to L :m_\alpha (x) = \alpha x, is a ''K''-linear transformation of this vector space into itself. The norm, N''L''/''K''(''α''), is defined as the determinant of this linear transformation. If ''L''/''K'' is a Galois extension, one may compute the norm of ''α'' ∈ ''L'' as the product of all the Galois conjugates of ''α'': :\operatorname_(\alpha)=\prod_ \sigma(\alpha), where Gal(''L''/''K'') denotes the Galois group of ''L''/''K''. (Note that there may be a repetition in the terms of the product.) For a general field extension ''L''/''K'', and nonzero ''α'' in ''L'', let ''σ''(''α ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasifield
In mathematics, a quasifield is an algebraic structure (Q,+,\cdot) where + and \cdot are binary operations on Q, much like a division ring, but with some weaker conditions. All division rings, and thus all fields, are quasifields. Definition A quasifield (Q,+,\cdot) is a structure, where + and \cdot are binary operations on Q, satisfying these axioms: * (Q,+) is a group * (Q_,\cdot) is a loop, where Q_ = Q \setminus \ \, * a \cdot (b+c)=a \cdot b+a \cdot c \quad\forall a,b,c \in Q (left distributivity) * a \cdot x=b \cdot x+c has exactly one solution for x, \forall a,b,c \in Q, a\neq b Strictly speaking, this is the definition of a ''left'' quasifield. A ''right'' quasifield is similarly defined, but satisfies right distributivity instead. A quasifield satisfying both distributive laws is called a semifield, in the sense in which the term is used in projective geometry. Although not assumed, one can prove that the axioms imply that the additive group (Q,+) is abelian. Thus, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation Plane
In mathematics, a translation plane is a projective plane which admits a certain group of symmetries (described below). Along with the Hughes planes and the Figueroa planes, translation planes are among the most well-studied of the known non-Desarguesian planes, and the vast majority of known non-Desarguesian planes are either translation planes, or can be obtained from a translation plane via successive iterations of Duality (projective geometry), dualization and/or Hall plane#Derivation, derivation. In a projective plane, let represent a point, and represent a line. A ''Homography#Central collineations, central collineation'' with ''center'' and ''axis'' is a collineation fixing every point on and every line through . It is called an ''elation'' if is on , otherwise it is called a ''homology''. The central collineations with center and axis form a group. A line in a projective plane is a ''translation line'' if the group of all elations with axis Group action (mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
