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Just as in chemistry, the characteristic property of a material will serve to identify a sample, or in the study of materials, structures and properties will determine characterization, in mathematics there is a continual effort to express properties that will distinguish a desired feature in a theory or system. Characterization is not unique ...
A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
For example, if p is prime and q(X) is an irreducible polynomial with coefficients in the field with p elements, then the quotient ring [] / (()) is a field of characteristic p. Another example: The field C {\displaystyle \mathbb {C} } of complex numbers contains Z {\displaystyle \mathbb {Z} } , so the characteristic of C {\displaystyle \mathbb ...
Non-Archimedean local fields of characteristic p (for p any given prime number): the field of formal Laurent series F q ((T)) over a finite field F q, where q is a power of p. In particular, of importance in number theory, classes of local fields show up as the completions of algebraic number fields with respect to their discrete valuation ...
Purely inseparable extensions do occur naturally; for example, they occur in algebraic geometry over fields of prime characteristic. If K is a field of characteristic p , and if V is an algebraic variety over K of dimension greater than zero, the function field K ( V ) is a purely inseparable extension over the subfield K ( V ) p of p th powers ...
In mathematics, the characteristic equation (or auxiliary equation [1]) is an algebraic equation of degree n upon which depends the solution of a given n th-order differential equation [2] or difference equation. [3] [4] The characteristic equation can only be formed when the differential equation is linear and homogeneous, and has constant ...
In the example above, the discriminant of the number field () with x 3 − x − 1 = 0 is −23, and as we have seen the 23-adic place ramifies. The Dedekind discriminant tells us it is the only ultrametric place that does.
then F is said to have characteristic 0. [11] For example, the field of rational numbers Q has characteristic 0 since no positive integer n is zero. Otherwise, if there is a positive integer n satisfying this equation, the smallest such positive integer can be shown to be a prime number.