Reduced Ramification Index Of An Extension Of Valuations
In algebra (in particular in algebraic geometry or algebraic number field, algebraic number theory), a valuation is a Function (mathematics), function on a Field (mathematics), field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole (complex analysis), pole or Multiplicity (mathematics), multiplicity of a zero (complex analysis), zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact (geometry), contact between two algebraic variety, algebraic or analytic variety, analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field. Definition One starts with the following objects: *a field (mathematics), field and its multiplicative group ''K''×, *an abelian group, abelian totally ordered group . The ordering and group law on are ex ...
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Algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication. Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called '' systems of linear equations''. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. Abstract algebra studies algebraic structures, which consist of a set of mathemati ...
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