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29 relations: Abelian extension, Algebraic integer, Algebraic number field, Algebraic number theory, Carl Friedrich Gauss, Characteristic (algebra), Chebotarev's density theorem, Conjugacy class, Conjugate element (field theory), Coprime integers, Cyclotomic field, Dirichlet's theorem on arithmetic progressions, Euler's totient function, Ferdinand Georg Frobenius, Frobenius endomorphism, Galois extension, Galois group, Gaussian integer, Ideal number, Klein four-group, Natural density, Nikolai Chebotaryov, Partition (number theory), Permutation, Prime number, Rational number, Siegel zero, Splitting field, Symmetric group.
Abelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian.
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Algebraic integer
In algebraic number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in (the set of integers).
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Algebraic number field
In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
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Algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations.
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Carl Friedrich Gauss
Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.
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Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.
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Chebotarev's density theorem
Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field \mathbb of rational numbers.
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Conjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure.
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Conjugate element (field theory)
In mathematics, in particular field theory, the conjugate elements of an algebraic element α, over a field extension L/K, are the roots of the minimal polynomial pK,α(x) of α over K. Conjugate elements are also called Galois conjugates or simply conjugates.
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Coprime integers
In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.
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Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to, the field of rational numbers.
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Dirichlet's theorem on arithmetic progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is a non-negative integer.
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Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to.
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Ferdinand Georg Frobenius
Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory.
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Frobenius endomorphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic, an important class which includes finite fields.
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Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.
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Galois group
In mathematics, more specifically in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.
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Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers.
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Ideal number
In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings.
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Klein four-group
In mathematics, the Klein four-group (or just Klein group or Vierergruppe, four-group, often symbolized by the letter V or as K4) is the group, the direct product of two copies of the cyclic group of order 2.
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Natural density
In number theory, natural density (or asymptotic density or arithmetic density) is one of the possibilities to measure how large a subset of the set of natural numbers is.
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Nikolai Chebotaryov
Nikolai Grigorievich Chebotaryov (often spelled Chebotarov or Chebotarev, Никола́й Григо́рьевич Чеботарёв, Микола Григорович Чоботарьов) (– 2 July 1947) was a noted Russian and Soviet mathematician.
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Partition (number theory)
In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers.
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Permutation
In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting.
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Prime number
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
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Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
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Siegel zero
In mathematics, more specifically in the field of analytic number theory, a Siegel zero, named after Carl Ludwig Siegel, is a type of potential counterexample to the generalized Riemann hypothesis, on the zeroes of Dirichlet L-function.
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Splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial splits or decomposes into linear factors.
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Symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.
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Redirects here:
Chebotarev density theorem, Chebotaryov density theorem, Chebotaryov's density theorem, Tchebotarev density theorem.
References
[1] https://en.wikipedia.org/wiki/Chebotarev's_density_theorem
