|
Steinitz's Theorem (field Theory)
In field theory, Steinitz's theorem states that a finite extension of fields L/K is simple if and only if there are only finitely many intermediate fields between K and L. Proof Suppose first that L/K is simple, that is to say L = K(\alpha) for some \alpha \in L. Let M be any intermediate field between L and K, and let g be the minimal polynomial of \alpha over M. Let M' be the field extension of K generated by all the coefficients of g. Then M' \subseteq M by definition of the minimal polynomial, but the degree of L over M' is (like that of L over M) simply the degree of g. Therefore, by multiplicativity of degree, :M'= 1 and hence M = M'. But if f is the minimal polynomial of \alpha over K, then g , f, and since there are only finitely many divisors of f, the first direction follows. Conversely, if the number of intermediate fields between L and K is finite, we distinguish two cases: #If K is finite, then so is L, and any primitive root of L will generate the field extensi ... [...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] |
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 'E'':''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(' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Extension
In field theory, a simple extension is a field extension that is generated by the adjunction of a single element, called a ''primitive element''. Simple extensions are well understood and can be completely classified. The primitive element theorem provides a characterization of the finite simple extensions. Definition A field extension is called a simple extension if there exists an element in ''L'' with :L = K(\theta). This means that every element of can be expressed as a rational fraction in , with coefficients in ; that is, it is produced from and elements of by the field operations +, −, •, / . Equivalently, is the smallest field that contains both ' and . There are two different kinds of simple extensions (see below): # The element may be transcendental over , which means that it is not a root of any polynomial with coefficients in . In this case K(\theta) is isomorphic to the field of rational functions K(X). # Otherwise, is algebraic over ; that is, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Polynomial (field Theory)
In field theory, a branch of mathematics, the minimal polynomial of an element of an extension field of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the smaller field, such that is a root of the polynomial. If the minimal polynomial of exists, it is unique. The coefficient of the highest-degree term in the polynomial is required to be 1. More formally, a minimal polynomial is defined relative to a field extension and an element of the extension field . The minimal polynomial of an element, if it exists, is a member of , the ring of polynomials in the variable with coefficients in . Given an element of , let be the set of all polynomials in such that . The element is called a root or zero of each polynomial in More specifically, ''J''''α'' is the kernel of the ring homomorphism from ''F'' 'x''to ''E'' which sends polynomials ''g'' to their value ''g''(''α'') at the element ''α''. Because it is the kernel of a ring homom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stacks Project
The Stacks Project is an open source collaborative mathematics textbook writing project with the aim to cover "algebraic stacks and the algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ... needed to define them". , the book consists of 116 chapters (excluding the license and index chapters) spreading over 7500 pages. The maintainer of the project, who reviews and accepts the changes, is Aise Johan de Jong. References External linksProject website*Latest from the Stacks Project(as of 2013) (Accessed 1 April 2020)Kerodona Stacks project inspired online textbook on categorical homotopy theory maintained by Jacob Lurie Mathematics textbooks {{mathematics-lit-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Steinitz
Ernst Steinitz (13 June 1871 – 29 September 1928) was a German mathematician. Biography Steinitz was born in Laurahütte ( Siemianowice Śląskie), Silesia, Germany (now in Poland), the son of Sigismund Steinitz, a Jewish coal merchant, and his wife Auguste Cohen; he had two brothers. He studied at the University of Breslau and the University of Berlin, receiving his Ph.D. from Breslau in 1894. Subsequently, he took positions at Charlottenburg (now Technische Universität Berlin), Breslau, and the University of Kiel, Germany, where he died in 1928. Steinitz married Martha Steinitz and had one son. Mathematical works Steinitz's 1894 thesis was on the subject of projective configurations; it contained the result that any abstract description of an incidence structure of three lines per point and three points per line could be realized as a configuration of straight lines in the Euclidean plane with the possible exception of one of the lines. His thesis also contains th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
