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{{r|Algebraic number}}

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In [[mathematics]], and more specifically—in [[number theory]], an '''algebraic number''' is a [[complex number]] that is a root of a [[polynomial]] with [[ration ...s that ensued forms the foundation of modern [[algebraic number theory]]. Algebraic number theory is now an immense field, and one of current research, but so far has

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{{r|Discriminant of an algebraic number field}}

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...e of integral closure is the [[ring of integers]] or maximal order in an [[algebraic number field]] ''K'', which may be defined as the integral closure of '''Z''' in ' * {{cite book | author=Pierre Samuel | authorlink=Pierre Samuel | title=Algebraic number theory | publisher=Hermann/Kershaw | year=1972 }}

...], the '''different ideal''' is an invariant attached to an extension of [[algebraic number field]]s. ...tive norm]] of the relative different is equal to the [[Discriminant of an algebraic number field|relative discriminant]] Δ_''L''/''K''. In a tower of fiel

...of a [[square matrix]], an [[endomorphism]] of a [[vector space]] or an [[algebraic number]]. ==Minimal polynomial of an algebraic number==

...'''cycle''') is a formal product of [[Place (mathematics)|place]]s of an [[algebraic number field]]. It is used to encode [[ramification]] data for [[abelian extensio Let ''K'' be an algebraic number field with ring of integers ''R''. A ''modulus'' is a formal product

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In [[mathematics]], to each [[algebraic number field]] ''k'', there is associated an important function called the '''Dede If ''k'' is an algebraic number field, the Dedekind zeta function of the field is a [[meromorphic function]

...ory]], an '''algebraic number field''' is a principal object of study in [[algebraic number theory]]. The algebraic and arithmetic structure of a number field has app ...field]] '''Q''' of [[rational number]]s. The elements of ''K'' are thus [[algebraic number]]s. Let ''n'' = [''K'':'''Q'''] be the degree of the extension.

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...thor=A. Fröhlich | authorlink=Ali Fröhlich | coauthors=M.J. Taylor | title=Algebraic number theory | series=Cambridge studies in advanced mathematics | volume=27 | pub * {{cite book | author=Pierre Samuel | authorlink=Pierre Samuel | title=Algebraic number theory | publisher=Hermann/Kershaw | year=1972 }}

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...thor=A. Fröhlich | authorlink=Ali Fröhlich | coauthors=M.J. Taylor | title=Algebraic number theory | series=Cambridge studies in advanced mathematics | volume=27 | pub ...art | authorlink=Ian Stewart (mathematician) | coauthors=D.O. Tall | title=Algebraic number theory | publisher=Chapman and Hall | year=1979 | isbn=0-412-13840-9 | page

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In [[mathematics]], in the field of [[algebraic number theory]], an '''''S'''''<nowiki></nowiki>'''-unit''' generalises the idea o *{{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebraic number theory | publisher=Springer | isbn=0-387-94225-4 | year=1986 }} Chap. V.

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...athbb{Q}</math>, i.e. the field of roots of rational polynomials, is the [[algebraic number]]s.

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The term '''rational integer''' is used in [[algebraic number theory]] to distinguish these "ordinary" integers, embedded in the [[field

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...These include the [[unique factorization|unique factorization theorem]], [[algebraic number fields]], [[elliptic curves]], and [[modular form]]s.

...pace, there is an associated Artin L-function. When ''K'' and ''k'' are [[algebraic number field]]s, Artin L-functions generalize [[Dedekind zeta function]]s, which a

...algebraic structures, such as algebraic topology, algebraic geometry, and algebraic number theory. A strong understanding of module theory is essential for anyone de

#Let <math>K</math> be an [[algebraic number field]]. Then the integral closure <math>o_K</math>of <math>\mathbb{Z}</mat

...the study of [[polynomial|polynomial rings]] and [[Algebraic number field|algebraic number fields]] in the second half of the nineteenth century, amongst other by [[R

Let ''K'' be an [[algebraic number field]], a finite [[field extension|extension]] of '''Q''', and ''E'' an el

...[[rational number|rational]] or [[irrational number|irrational]]; either [[algebraic number|algebraic]] or [[transcendental number|transcendental]]; and either [[posit ...real numbers is [[uncountable|uncountably infinite]] but the set of all [[algebraic number]]s is [[countable|countably infinite]]. Contrary to widely held beliefs, hi

...odern subfields - in particular, [[analytic number theory|analytic]] and [[algebraic number theory]]. Algebraic number theory may be said to start with the study of reciprocity and cyclotomy, bu

...umber that is solution to a [[polynomial]] in integer coefficients is an [[algebraic number]]. This set includes all rational numbers and a subset of the irrational nu

...r, when one generalizes the concept of "integer", and starts considering [[algebraic number|algebraic integers]], it becomes clear that, while one can study elements t

...ed ideas of his teacher, Artin; some of the most interesting passages in ''Algebraic Number Theory'' also reflect Artin's influence and ideas that might otherwise not

...ay, Gauss arguably made a first foray towards both [[Galois]]'s work and [[algebraic number theory]]. ...its rough subdivision into its current subfields - especially analytic and algebraic number theory - dates from that period.