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Topic: Cyclotomic field

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	Quadratic field - Wikipedia, the free encyclopedia
		Such extensions run over all field extensions of the rational number field that are of degree 2 (quadratic extensions).
		A classical example of the construction of a quadratic field is to take the unique quadratic field inside the cyclotomic field generated by a primitive p-th root of unity, with p a prime number > 2.
		As explained at Gaussian period, the discriminant of the quadratic field is p for p = 4n + 1 and −p for p = 4n + 3.
	en.wikipedia.org /wiki/Quadratic_field (479 words)

	GAP Manual: 1.21 About Fields
		Cyclotomic fields are constructed as extensions of the Rationals by primitive roots of unity.
		For a cyclotomic a this is the smallest cyclotomic field that contains a (note that this is not the smallest field that contains a, which may be a number field that is not a cyclotomic field).
		As before with cyclotomic fields, the Galois group of a finite field and the norm and trace of its elements may be computed.
	www-groups.dcs.st-and.ac.uk /~gap/Gap3/Manual3/C001S021.htm (1208 words)

	U11 - Online Information article about U11
		field (n); and the prime factors of any number an+b, as well as the degree of their multiplicity, may be found by factorizing (6a2—ab+b2), the norm of (an+b).
		For a quadratic field the equation is of the form hie--nhz2 = =1, and the theory of this is complete; but except for certain special cubic corpora little has been done towards solving the important problem of assigning a definite process by which, for a given field, a system of fundamental units may be calculated.
		A cyclotomic field is one the elements of which are all expressible as rational functions of roots of unity; in particular the complete cyclotomic field Cm, of order 4,(m), is the aggregate of all rational functions of a primitive mth root of unity.
	encyclopedia.jrank.org /TUM_VAN/U11.html (13296 words)

	Algebraic number field - Wikipedia, the free encyclopedia
		In mathematics, an algebraic number field (or simply number field) is a finite (and therefore algebraic) field extension of the rational numbers Q.
		That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q.
		The study of algebraic number fields, and these days also of infinite algebraic extensions of the rational number field, is the central topic of algebraic number theory.
	en.wikipedia.org /wiki/Algebraic_number_field (105 words)

	[ref] 18 Cyclotomic Numbers
		Cyclotomics are usually entered as sums of roots of unity, with rational coefficients, and irrational cyclotomics are displayed in the same way.
		Since the underlying basis of the external representation of cyclotomics is an integral basis (see Integral Bases for Abelian Number Fields), the subring of cyclotomic integers in a cyclotomic field is formed by those cyclotomics for which the external representation is a list of integers.
		A special integral basis of cyclotomic fields is chosen that allows one to easily convert arbitrary sums of roots of unity into the basis, as well as to convert a cyclotomic represented w.r.t.
	www.math.sunysb.edu /~sorin/online-docs/gap4r3/htm/ref/CHAP018.htm (2008 words)

	PlanetMath: algebraic number theory (Site not responding. Last check: 2007-10-08)
		The cyclotomic units are a subgroup of the group of units of a cyclotomic field with very interesting properties.
		The Dedekind zeta function of a number field satisfies the so-called class number formula, which relates many of the invariants of the number field.
		Class field theory and the Artin map can be presented in terms of idèles and adèles.
	planetmath.org /encyclopedia/AlgebraicNumberTheory.html (936 words)

	GAP Manual: 13.1. More about Cyclotomics (Site not responding. Last check: 2007-10-08)
		Elements of number fields (see chapter Subfields of Cyclotomic Fields), cyclotomics for short, are arithmetical objects like rationals and finite field elements; they are not implemented as records ---like groups--- or e.g.
		Cyclotomics are usually entered as (and irrational cyclotomics are always displayed as) sums of roots of unity with rational coefficients.
		Cyclotomics are always represented in the smallest cyclotomic field they are contained in.
	www.math.uiuc.edu /Software/GAP-Manual/More_about_Cyclotomics.html (289 words)

	GAP Manual: 15 Subfields of Cyclotomic Fields
		Thus number fields are the domains (see chapter Domains) related to cyclotomics; they are special field records (see Field Records) which are needed to specify the field extension with respect to which e.g.
		In many situations cyclotomic fields need not be treated in a special way, except that there may be more efficient algorithms for them than for arbitrary number fields.
		The Galois automorphisms of the cyclotomic field Q_n are given by linear extension of the maps ast k: e_n mapsto e_n^k with 1 leq k
	www.mcs.kent.edu /system/documentation/gap/CHAP015.htm (1550 words)

	Creation Functions
		Given an element a from a cyclotomic field F, this function returns the smallest cyclotomic field (possibly the rational field) E subset F containing a.
		Given a sequence of cyclotomic field elements s, this function returns the smallest cyclotomic field (possibly the rational field) G containing each of those.
		Given a cyclotomic field Q(zeta_m) and an integer n>2, create the n-th root of unity zeta_n in K. An error results if zeta_n notin K, that is, if n does not divide m (or 2m in case m is odd).
	www.math.uiuc.edu /Software/magma/text350.html (523 words)

	GAP Manual: 6 Fields
		Fields are domains, so all functions that are applicable to all domains are also applicable to fields (see chapter Domains).
		A field homomorphism phi is a mapping that maps each element of a field F, called the source of phi, to an element of another field G, called the range of phi, such that for each pair x,y in F we have (x+y)^phi = x^phi + y^phi and (xy)^phi = x^phi y^phi.
		Since field homomorphisms are just a special case of homomorphisms, all functions described in chapter Homomorphisms are applicable to all field homomorphisms, e.g., the function to test if a homomorphism is a an automorphism (see IsAutomorphism).
	www.mcs.kent.edu /system/documentation/gap/CHAP006.htm (2478 words)

	Root of unity -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		The n-th roots of unity are precisely the zeros of the (A mathematical expression that is the sum of a number of terms) polynomial p(X) = X
		This (A piece of land cleared of trees and usually enclosed) field contains all nth roots of unity and is the (Click link for more info and facts about splitting field) splitting field of the nth cyclotomic polynomial over Q.
		Conversely, every abelian extension of the rationals is such a subfield of a cyclotomic field - a theorem of (German mathematician (1823-1891)) Kronecker, usually called the (Click link for more info and facts about Kronecker-Weber theorem) Kronecker-Weber theorem on the grounds that Weber supplied the proof.
	www.absoluteastronomy.com /encyclopedia/r/ro/root_of_unity.htm (822 words)

	Gaussian period - Wikipedia, the free encyclopedia
		In mathematics, a Gaussian period is a certain kind of sum of roots of unity.
		They permit explicit calculations in cyclotomic fields, in relation both with Galois theory and with harmonic analysis (discrete Fourier transform).
		The Gaussian periods lie in smaller fields, in general, since the values of the χ(a) when n is a prime p are (p − 1)-th roots of unity.
	www.wikipedia.org /wiki/Gauss_sum (733 words)

	P-adic number - Wikipedia, the free encyclopedia
		For example, the field of p-adic analysis essentially provides an alternative form of calculus.
		of p-adic numbers is an extension field of the rational numbers.
		The partial sums of this latter series are the elements of the given sequence.
	en.wikipedia.org /wiki/P-adic_number (2053 words)

	Operations on Structures
		The number of elements in the local ring or field L. The cardinality is finite only if L is a quotient ring or a bounded free precision ring.
		Given a local ring or field L and an integer k, return the generator of L if k is 1; otherwise, raise an error.
		Given a local ring or field L and a non-negative single precision integer k, change the maximum precision with which elements can be created to be k.
	www.math.lsu.edu /magma/text749.htm (394 words)

	Connections between Cubic and Dual Quadratic Fields
		Since discriminants of quadratic fields are essentially squarefree (possibly up to their 2-power contribution), one of two dual discriminants is divisible by 3, the other is not.
		The collection of all quadratic number fields is the disjoint union of all dual pairs up to a single exception.
		class field theory [2] (the ARTIN correspondence between subgroups of the 3-elementary ideal class group of k and unramified cyclic cubic extensions of k, resp.
	www.algebra.at /mirror.htm (801 words)

	Operations on Elements
		Given a local ring or field element x, return the norm of x over the local ring or field R. The ring R must be a subring of the parent of x.
		Given a local ring or field element x, return the trace of x over the local ring or field R. The ring R must be a subring of the parent of x.
		Given a local ring or field element x, return the minimal polynomial of x over the local ring or field R. The ring R must be a subring of the parent of x.
	www.math.lsu.edu /magma/text751.htm (2000 words)

	GAP Manual: 13 Cyclotomics (Site not responding. Last check: 2007-10-08)
		This is because it is easy to embed two cyclotomic fields in a larger one that contains both, i.e., there is a natural way to get the sum or the product of two arbitrary cyclotomics as element of a cyclotomic field.
		Since the base used is an integral base (see ZumbroichBase), the subring of cyclotomic integers in a cyclotomic field is formed by those cyclotomics which have not only rational but integral coefficients in their representation as sums of roots of unity.
		Cyclotomics are ordered as follows: The relation between rationals is as usual, and rationals are smaller than irrational cyclotomics.
	www.math.jussieu.fr /~jmichel/htm/CHAP013.htm (1660 words)

	Imprimitive degree 9 fields
		In this example we illustrate how to construct some of the fields that are the subject of study in a certain paper, and show how to verify some of the results mentioned there.
		The field L is one of 3 imprimitive degree 9 fields given in Diaz y Diaz and Olivier for which the class group is
		In Diaz y Diaz and Olivier's paper, 4 non-isomorphic fields are constructed that are all relative cubic extensions of discriminant -2045563163 of the cubic field k of discriminant 49 defined above.
	magma.maths.usyd.edu.au /magma/Examples/node61.html (502 words)

	Creation Functions
		Given a positive integer p, create and return a version C of the complex field C in which all calculations are correct to p' decimal digits, where p'=4.Ceiling((p/4)) is the smallest multiple of 4 greater than or equal to p.
		If the default field is the free real field, in the absence of p the maximum of the default precision and the number of decimal digits supplied is used in the creation of the free real number r.
		If R is a free real field and a is a fixed precision real number again an element in R approximating a as well as possible to the default precision of R is returned.
	www.umich.edu /~gpcc/scs/magma/text566.htm (801 words)

	Creation Functions (Site not responding. Last check: 2007-10-08)
		Nevertheless, it may be necessary to formally create the rational field, for instance if it is to be used as the coefficient ring for a polynomial ring.
		Given the rational field Q, and an integer a, create the rational number a=a/1 in Q. Also, any element from a quadratic, cyclotomic or number field (or an order of such) that is rational can be coerced into the rational field this way.
		This function returns, in general, for a positive integer n and a cyclotomic field Q a primitive n-th root of unity in Q; if Q is the rational field, n must be 1 or 2, and the result will be 1 or -1 in Q accordingly.
	www.math.lsu.edu /magma/text537.htm (327 words)

	Cyclotomic field (Site not responding. Last check: 2007-10-08)
		− 1; the primitive n-th roots of unity are precisely thezeros of the nth cyclotomic polynomial
		Every subfield of a cyclotomic field is an abelianextension of the rationals.
		Conversely, every abelian extension of the rationals is such a subfield of a cyclotomic field - a theorem of Kronecker, usually called the Kronecker-Weber theorem on thegrounds that Weber supplied the proof.
	www.therfcc.org /cyclotomic-field-218234.html (532 words)

	ABSTRACT ALGEBRA ON LINE: Galois Theory
		To study solvability by radicals of a polynomial equation f(x) = 0, we let K be the field generated by the coefficients of f(x), and let F be a splitting field for f(x) over K. Galois considered permutations of the roots that leave the coefficient field fixed.
		Let K be a field of characteristic zero, and let E be a radical extension of K. Then there exists an extension F of E that is a normal radical extension of K. Theorem.
		A set that satisfies all the axioms of a field except for commutativity of multiplication is called a division ring or skew field.
	www.math.niu.edu /~beachy/aaol/galois.html (1898 words)

	Root of unity : Cyclotomic field (Site not responding. Last check: 2007-10-08)
		By adjoining a primitive n-th root of unity to Q, one obtains the n-th cyclotomic field F
		This field contains all n-th roots of unity and is the splitting field of the n-th cyclotomic polynomial over Q.
		Conversely, every abelian extension of the rationals is such a subfield of a cyclotomic field - a theorem of Kronecker[?].
	www.termsdefined.net /cy/cyclotomic-field.html (695 words)

	[No title]
		] The extension field of a given field K which is the smallest extension field of K that includes the nth roots of unity for some integer n.
		] An accelerator in which charged particles are successively accelerated by a constant-frequency alternating electric field that is synchronized with movement of the particles on spiral paths in a constant magnetic field normal to their path.
		] Resonance absorption of energy from an alternating-current electric field by electrons or ions in a uniform magnetic field when the frequency of the electric field equals the cyclotron frequency, or the cyclotron frequency corresponds to the effective mass of electrons in a solid.
	www.accessscience.com /Dictionary/C/C65/DictC65.html (2243 words)

	Structure Operations (Site not responding. Last check: 2007-10-08)
		In cyclotomic fields the generic ring functions are supported.
		The vector space isomorphic to the cyclotomic field K as a vector space over J and the isomorphism from K to the vector space.
		The smallest n such that the field K is contained in Q(zeta_n); for a cyclotomic field that is either the `cyclotomic order' m (see below) or half that, depending on whether m = 2 mod 4.
	www.umich.edu /~gpcc/scs/magma/text655.htm (219 words)

	Algebraic numbers
		On the pages about Quadratic fields and Cyclotomic fields I showed how you could extend the rational to larger number fields.
		So the numbers in quadratic fields and cyclotomic fields are algerbraic numbers according to this definition.
		So such a relation exists for the powers of an element of this field from the 0-th power (1) to the n-th power, so it is indeed algebraic.
	homepages.cwi.nl /~dik/english/mathematics/numf.html (996 words)

	GAP Manual: 15 Subfields of Cyclotomic Fields (Site not responding. Last check: 2007-10-08)
		The only number fields that GAP can handle at the moment are subfields of cyclotomic fields, e.g., Q(√5) is a number field that is not cyclotomic but contained in the cyclotomic field Q
		is an extension of a number field that is not cyclotomic; used by Coefficients
		The only possibility where it is allowed to prescribe a base is when the field is constructed (see Number Field Records, Cyclotomic Field Records).
	www.math.jussieu.fr /~jmichel/htm/CHAP015.htm (1465 words)

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