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The Regents Examinations are developed and administered by the New York State Education Department (NYSED) under the authority of the Board of Regents of the University of the State of New York. Regents exams are prepared by a conference of selected New York teachers of each test's specific discipline who assemble a test map that highlights the ...
An extension is unramified if, and only if, the discriminant is the unit ideal. The Minkowski bound above shows that there are no non-trivial unramified extensions of Q. Fields larger than Q may have unramified extensions: for example, for any field with class number greater than one, its Hilbert class field is a non-trivial unramified extension.
GeoGebra is software that combines geometry, algebra and calculus for mathematics education in schools and universities. It is available free of charge for non-commercial users. [6] License: open source under GPL license (free of charge) Languages: 55. Geometry: points, lines, all conic sections, vectors, parametric curves, locus lines.
Finite extensions of local fields. In algebraic number theory, through completion, the study of ramification of a prime ideal can often be reduced to the case of local fields where a more detailed analysis can be carried out with the aid of tools such as ramification groups . In this article, a local field is non-archimedean and has finite ...
In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of n th roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's Last Theorem. The main statements do not depend on ...
Ramification in algebraic number theory means a prime ideal factoring in an extension so as to give some repeated prime ideal factors. Namely, let be the ring of integers of an algebraic number field , and a prime ideal of . For a field extension we can consider the ring of integers (which is the integral closure of in ), and the ideal of .
Degree of a field extension. In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory —indeed in any area where fields appear prominently.
Étale morphism. In algebraic geometry, an étale morphism ( French: [etal]) is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology.