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Exponential Ring
In mathematics, an exponential field is a Field (mathematics), field that has an extra operation on its elements which extends the usual idea of exponentiation. Definition A field is an algebraic structure composed of a set of elements, ''F'', two binary operations, addition (+) such that ''F'' forms an abelian group with identity 0''F'' and multiplication (ยท), such that ''F'' excluding 0''F'' forms an abelian group under multiplication with identity 1''F'', and such that multiplication is distributive over addition, that is for any elements ''a'', ''b'', ''c'' in ''F'', one has . If there is also a Function (mathematics), function ''E'' that maps ''F'' into ''F'', and such that for every ''a'' and ''b'' in ''F'' one has :\begin&E(a+b)=E(a)\cdot E(b),\\&E(0_F)=1_F \end then ''F'' is called an exponential field, and the function ''E'' is called an exponential function on ''F''. Thus an exponential function on a field is a homomorphism between the additive group of ''F'' and its m ... [...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] |
