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Rabinowitsch Trick
In mathematics, the Rabinowitsch trick, introduced by George Yuri Rainich and published under his original name , is a short way of proving the general case of the Hilbert Nullstellensatz from an easier special case (the so-called ''weak'' Nullstellensatz), by introducing an extra variable. The Rabinowitsch trick goes as follows. Let ''K'' be an algebraically closed field. Suppose the polynomial ''f'' in ''K'' 'x''1,...''x''''n''vanishes whenever all polynomials ''f''1,....,''f''''m'' vanish. Then the polynomials ''f''1,....,''f''''m'', 1 − ''x''0''f'' have no common zeros (where we have introduced a new variable ''x''0), so by the weak Nullstellensatz for ''K'' 'x''0, ..., ''x''''n''they generate the unit ideal of ''K'' 'x''0 ,..., ''x''''n'' Spelt out, this means there are polynomials g_0,g_1,\dots,g_m \in K _0,x_1,\dots,x_n/math> such that : 1 = g_0(x_0,x_1,\dots,x_n) (1 - x_0 f(x_1,\dots,x_n)) + \sum_^m g_i(x_0,x_1,\dots,x_n) f_i(x_1,\dots,x_n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Yuri Rainich
George Yuri Rainich (Rabinovich) (March 25, 1886 in Odessa – October 10, 1968) was a leading mathematical physicist in the early twentieth century. Career Rainich studied mathematics from 1904 to 1908 in Odessa, in Göttingen (1905–1906), and in Munich (1906–1907), eventually obtaining his doctorate (Magister of Pure Mathematics) in 1913 from the University of Kazan. After teaching at the University of Kazan, in 1922 (via Istanbul), he emigrated with his wife to the United States. After three years at Johns Hopkins University, he joined the faculty of the University of Michigan, where he remained until his retirement in 1956. After his retirement as professor emeritus, he was in 1957 at Brown University as a member of the editorial staff of ''Mathematical Reviews'' and he was for several years a visiting professor at the University of Notre Dame. After the death of his wife in 1963, he returned to the University of Michigan at Ann Arbor and organized there a seminar on gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Nullstellensatz
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert, who proved the Nullstellensatz in his second major paper on invariant theory in 1893 (following his seminal 1890 paper in which he proved Hilbert's basis theorem). Formulation Let ''k'' be a field (such as the rational numbers) and ''K'' be an algebraically closed field extension (such as the complex numbers). Consider the polynomial ring k _1, \ldots, X_n/math> and let ''I'' be an ideal in this ring. The algebraic set V(''I'') defined by this ideal consists of all ''n''-tuples x = (''x''1,...,''x''''n'') in ''Kn'' such that ''f''(x) = 0 for all ''f'' in ''I''. Hilbert's Nullstellensatz st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraically Closed Field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because the polynomial equation ''x''2 + 1 = 0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed. Also, no finite field ''F'' is algebraically closed, because if ''a''1, ''a''2, ..., ''an'' are the elements of ''F'', then the polynomial (''x'' − ''a''1)(''x'' − ''a''2) ⋯ (''x'' − ''a''''n'') + 1 has no zero in ''F''. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraicall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' join ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Of Fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral domain elements. The field of fractions of R is sometimes denoted by \operatorname(R) or \operatorname(R), and the construction is sometimes also called the fraction field, field of quotients, or quotient field of R. All four are in common usage, but are not to be confused with the quotient of a ring by an ideal, which is a quite different concept. For a commutative ring which is not an integral domain, the analogous construction is called the localization or ring of quotients. Definition Given an integral domain and letting R^* = R \setminus \, we define an equivalence relation on R \times R^* by letting (n,d) \sim (m,b) whenever nb = md. We denote the equivale ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Math
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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
