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==Primitive root of unity== ...e element $\zeta$ generates the [[cyclic group]] $\mu_m$ of roots of unity of order $m$.

An $n$-th root of a number $a$ is a number $x=a^{1/n}$ whose $n$-th power $x^n$ is equal to $ A root of an algebraic equation over a field $k$,

A function that associates with an ordered pair of elements $x,y$ of the multiplicative group $K^*$ of a

...(Cf. also [[Algebraic number]]; [[Algebraic number theory]]; [[Extension of a field]]; [[Number field]].) * [[Quadratic field]] — an extension of degree $n=2$;

...oots of unity). Every element $a$ of order $n$ can be taken as a generator of this group. Then

A field with a finite number of elements. First considered by E. Galois [[#References|[1]]]. ...(p)$ with respect to inclusion. The lattice of finite algebraic extensions of any Galois field within its fixed algebraic closure is such a lattice.

A resolvent of an algebraic equation $ f( x) = 0 $ of degree $ n $

''of a projective algebraic variety $X$ over a field $k$'' ...References|[1]]]). Thus, this Fano variety is reducible and each component of it is not reduced at a generic point.

...braic integer which is not a root of unity. The Mahler measure $M(\alpha)$ of an [[algebraic number]] $\alpha$ is defined by ...ds only on $f$, it is also denoted by $M(f)$ and called the Mahler measure of $f$. Jensen's formula (cf. also [[Jensen formula]]) implies the equality

...~/encyclopedia/old_files/data/F040/F.0400280 Finite group, representation of a A homomorphism of a finite group $ G $

...$m$. If this congruence is not solvable, then $a$ is called a non-residue of degree $n$ modulo $m$. When $n=2$, the power residues and non-residues are In the case of a prime modulus $p$, the question of the solvability of the congruence $x^n \equiv a \pmod p$ can be answered by using the [[Euler

objects on the basis of their automorphism groups. For instance, Galois theories of fields, rings, topological spaces, etc., are

Out of 37 formulas, 36 were replaced by TEX code.--> ...number|Algebraic number]]; [[Number field|Number field]]) or a completion of such an $F$ with regard to an (infinite or finite) prime.

Out of 90 formulas, 90 were replaced by TEX code.--> ...near space]] over a [[Field|field]] $\mathbf{F}$. Let $L ( X )$ be the set of all linear operators with domains and ranges in $X$ and let

...e in a suitable divisible group. If the equations stated in the definition of a divisible group have a unique solution, the group is called a $D$-group. ...polov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)</TD></TR></table>

of the number $ n $, itself. Over the field of complex numbers one has

An [[extension of a field]] $k$ of characteristic $p \ge 0$, of the type ...of the type $\mathbf{Q}(\zeta_n,a^{1/n})$, where $\mathbf{Q}$ is the field of rational numbers and $a \in \mathbf{Q}$.

is of interest. The odd correlation of $ ( a _ {i} ) $ The goal is the design of large sets $ {\mathcal S} = \{ a ^ {( 1 ) } \dots a ^ {( m ) } \} $

...stability of a linear closed-loop system formulated in terms of properties of the open-loop system. ...p system has $k$, $0\leq k\leq n$, roots with positive real part and $n-k$ roots with negative real part. The Nyquist criterion is as follows: The closed-lo

A complex (sometimes, real) number that is a root of a polynomial ...s any positive integer, is an algebraic number of degree $n$, being a root of the irreducible polynomial $x^n-2$.

...ion by A.O. Gel'fond of Hilbert's seventh problem and from subsequent work of A. Baker. The Gel'fond–Baker method also incorporates ideas due to C.L. S 1) integer points on curves of genus $ 0 $

The Schur index of a central simple algebra $ A $ cf. [[Central simple algebra|Central simple algebra]]) is the degree of the division algebra $ D $

The ''Hasse invariant'' is an arithmetic invariant of various objects. The Hasse invariant $h(A)$ of a central simple algebra $A$ over a local

...rphic images of Chernikov groups are also Chernikov; further, an extension of a Chernikov group by a Chernikov group is again Chernikov. ...are two-generator infinite simple groups in which every proper subgroup is of prime order $p$.

Out of 131 formulas, 131 were replaced by TEX code.--> ...ts , n$, $i \neq j$, $a \in F$, let $x _ { i j } ( a )$ denote the element of $G$ which differs from the identity matrix only in the $( i , j )$-entry, w

...group of finite index in $\def\O{\mathcal{O}}G_\O$, where $\O$ is the ring of integers in an algebraic number field $k$ (cf. ...ng $c(G) = 1$. In its modern form, the congruence subgroup problem is that of computing the congruence subgroup kernel $c(G)$.

Out of 64 formulas, 64 were replaced by TEX code.--> ...solution over $\mathbf{F} [ T ]$, with $\mathbf{F}$ the algebraic closure of $\mathbf{F}$.

be the ring of integers of an algebraic number [[Field|field]] $ F $( which is also called the tame kernel of $ F $,

Out of 445 formulas, 443 were replaced by TEX code.--> ...roup this reduction is carried out with respect to the field of definition of the group.

is an integer, the power function is a particular case of a [[Rational function|rational function]]. When $ x $ is a [[Power|power]], and the properties of $ y = x ^ {a} $

By a [[modular form]] of weight $k$ one understands a function $f$ on the upper half-plane satisfyin is an element of some congruence subgroup of $\text{SL}(2,\mathbf{Z})$ (cf. also [[Modular function]]).

of order $ n $, is the transposed matrix of $ H $

of rational numbers by adjoining a primitive $ n $-th root of unity $ \zeta _ {n} $, is the field of rational $ p $-adic numbers. Since $ K _ {n} = K _ {2n} $

The notion of ''discriminant'' can have different meanings, depending on the context. ====Discriminant of a [[polynomial]]====

Out of 1 formulas, 1 were replaced by TEX code.--> ...s of characteristic zero is used for various purposes, mainly in the proof of the [[Lefschetz formula|Lefschetz formula]] and its application to zeta-fun

of vectors in a vector space $ V $ of the space $ V ^{*} $

on the set of integers that satisfies the following conditions: ...hmetic progressions. He developed the fundamental principles of the theory of Dirichlet characters [[#References|[2]]]–[[#References|[8]]], starting fr

is [[#References|[a5]]] the unique [[Galois extension|Galois extension]] of $ \mathbf Z/ {p ^ {m} } \mathbf Z $ of degree $ d $.

...f mathematics on the other, have determined the development of the concept of a number. ...provided by linguistics and ethnography. Primitive man clearly had no need of counting skills to determine whether or not a given collection was complete

Out of 186 formulas, 168 were replaced by TEX code.--> ...ups|Quantum groups]]) and a [[Super-group|super-group]], in which the role of Bose–Fermi statistics is now played by more general braid statistics. The

...ic number fields $ K $ of finite degree over the field $ \mathbf Q $ of rational numbers (cf. [[Algebraic number|Algebraic number]]). ...ots \omega _{n} ) $ ; this means that each algebraic integer (i.e. element of $ O _{K} $ ) can be written in the form $ x _{1} \omega _{1} + \dots +

''of a field $k$'' ...ras, while the element inverse to the class of an algebra $A$ is the class of its [[opposite algebra]]. Each non-zero class contains, up to isomorphism,

Out of 89 formulas, 86 were replaced by TEX code.--> ...ecture says that for a [[Finite group|finite group]] $G$ all torsion units of ${\bf Z} G$ are rationally conjugate to $\pm g$, $g \in G$:

A partial differential equation (system) of the form is a multi-index (a set of non-negative integers), $ | \alpha | = \sum {\alpha _ {i} } $,

...159 : ~/encyclopedia/old_files/data/B110/B.1100850 Brauer characterization of characters be a [[Finite group|finite group]]. A representation of $ G $

of differentiations such that the set of restrictions of the elements of $ \Delta $ coincides with the set of differentiations on $ F _{0} $.

$#C+1 = 230 : ~/encyclopedia/old_files/data/C023/C.0203110 Cohomology of algebras ...ule. This definition encompasses many cohomology theories of certain types of (universal) algebras.

Out of 157 formulas, 149 were replaced by TEX code.--> ...$. They are indexed by the set $P ^ { + } \subset \mathfrak { h } ^ { * }$ of dominant integral weights $\lambda$ relative to $\Delta ^ { + }$.

A process for obtaining a sequence of interpolation functions $ \{ f _ {n} ( z) \} $ of [[Interpolation|interpolation]] conditions. If the interpolation functions

of the base $ S $ of a family $ \{ X _{s} \} $

Out of 239 formulas, 233 were replaced by TEX code.--> ...ermat last theorem|Fermat last theorem]]). For details and generalizations of Iwasawa theory, see [[#References|[a10]]], [[#References|[a7]]], [[#Referen

...ier integral|Fourier integral]]). Classical harmonic analysis — the theory of Fourier series and Fourier integrals — underwent a rapid development, sti .../h0464201.png" /> be a complex-valued square-summable function on a circle of unit length (or on the segment <img align="absmiddle" border="0" src="https

A very flexible geometric construction aimed to represent a family of similar objects (''fibres'' or ''fibers'', depending on the preferred spell ...of a smooth manifold. The [[covering]]s are also a special particular form of a topological bundle (with discrete fibers).

...Lie type'' if it can be obtained as a central [[Quotient group|quotient]] of a finite reductive group. ...n(q))$, where $Z(\SL_n(q))$ is the centre of $\SL_n(q)$, is a finite group of Lie type called the projective special linear group. This group is a [[Simp

...tions (cf. [[Dirichlet L-function|Dirichlet $L$-function]]) form the basis of modern [[analytic number theory]]. In addition to Riemann's zeta-function o which converges absolutely and uniformly in any bounded domain of the complex $s$-plane for which $\sigma\geq1+\delta$, $\delta>0$. If $\sigm

[[Cohomology]] of [[algebraic variety]] of dimension $n$ over an algebraically closed field $k$ and let $\mathcal{L}$