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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.
In mathematics, a characteristic class is a way of associating to each principal bundle of X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" and whether it possesses sections .
Local class field theory in dimension one has its analogues in higher dimensions. The appropriate replacement for the multiplicative group becomes the nth Milnor K-group , where n is the dimension of the field, which then appears as the domain of a reciprocity map to the Galois group of the maximal abelian extension over the field.
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.
In mathematics, a field K is called a non-Archimedean local field if it is complete with respect to a metric induced by a discrete valuation v and if its residue field k is finite. [1] In general, a local field is a locally compact topological field with respect to a non-discrete topology . [ 2 ]
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 ...
In mathematics, the Chern–Simons forms are certain secondary characteristic classes. [1] The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from which the theory arose. [2]
For example, the field of rational numbers is a subfield of the real numbers, which is itself a subfield of the complex numbers. More generally, the field of rational numbers is (or is isomorphic to) a subfield of any field of characteristic. The characteristic of a subfield is the same as the characteristic of the larger field.
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