...very number field contains infinitely many elements. The field of rational numbers is contained in every number field.
...fixed complex number and $H(x)$ and $F(x)$ range over the polynomials with rational coefficients.
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...is representable in the form $a+b\sqrt{d}$, where $a$ and $b$ are rational numbers, $b\ne 0$, and $d$ is an integer which is not a perfect square. A real numb
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...attice of all points with integral rational coordinates on the plane. Such numbers were first considered in 1832 by [[Gauss, Carl Friedrich|C.F. Gauss]] in hi
...osed into a non-trivial product) of $\Gamma$ (the Gaussian primes) are the numbers of the form
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where $a$ and $b$ are real numbers, while $m$, $n$ and $p$ are rational numbers. The indefinite integral of a differential binomial,
is reduced to an integral of rational functions if at least one of the numbers $p$, $(m+1)/n$ and $p+(m+1)/n$ is an integer. In all other cases, the integ
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...field]] of constants or a finite extension of the field $\mathbb{Q}$ of [[rational number]]s (an [[algebraic number field]]).
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...hic to the projective line $\mathbf P^1$. A complete singular curve $X$ is rational if and only if its geometric genus $g$ is zero, that is, when there are no
...the field $\mathbf C$ of complex numbers, the (only) non-singular complete rational curve $X$ is the Riemann sphere $\mathbf C\cup\{\infty\}$.
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...positive rational number $-r''$: $-r'>-r''$. The absolute value $|r|$ of a rational number $r$ is defined in the usual way: $|r|=r$ if $r\geq0$ and $|r|=-r$ if
...s, they are uniquely determined by $r'$ and $r''$ themselves. The rational numbers form an [[Ordered field|ordered field]], denoted by $\mathbf Q$.
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where the minimum is over all pairs $h_0,h_1$ of integral rational numbers such that
...ber $\xi$ can be approximated by rational numbers. For all real irrational numbers one has
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...mple, the set of real numbers is uncountable, whereas that of the rational numbers is countable.
The uncountability of the set of real numbers is sometimes proved by the Cantor diagonalization principle (cf. [[Cantor t
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An element of an extension of the field of rational numbers (cf.
obtained by completing the field of rational numbers with respect to a
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A number that is not a rational number (i.e. an integer or a fraction). Geometrically, an irrational number
...tween any two numbers there is an irrational number. The set of irrational numbers is uncountable, is a set of the second category and has type $G_\delta$ (cf
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...The $p$-adic [[valuation]] (or order) $\nu_p({\cdot})$ on the field of [[rational number]]s is defined by $\nu(a/b) = r$ where $a,b$ are integers and $a/b =
...$p$-adic norm, and the $p$-adic valuation extends to the field of $p$-adic numbers.
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where $a$ and $b$ are real numbers and $m$, $n$ and $p$ are rational numbers, cannot be expressed in terms of elementary functions for any $m$, $n$ and
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...f the same [[cardinality]]. For example, the set of integers, the set of [[rational number]]s or the set of [[algebraic number]]s.
...nite: that is, a set of the same cardinality as some subset of the natural numbers.
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''of numbers''
...e basic arithmetic operations. Multiplication consists in assigning to two numbers $a,b$ (called the factors) a third number $c$ (called the product). Multipl
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An [[Abelian number field]] is an Abelian extension of the field of rational numbers.
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.../\mathbf{Z}_p$, where $\mathbf{Q}_p$ is the additive group of the field of rational [[P-adic number|$p$-adic number]]s and $\mathbf{Z}_p$ is the additive group
...numbers, and also maximal $p$-subgroups of the additive group of rational numbers modulo 1. The ring of endomorphisms of a group of type $p^\infty$ is isomor
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$#C+1 = 19 : ~/encyclopedia/old_files/data/E035/E.0305730 Entire rational function,
are real or complex numbers, and $ z $
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In the case $P=1$,$Q=1$ the [[Fibonacci numbers]] and [[Lucas numbers]] are the Lucas sequences of the first and second kind respectively.
...Lucas sequence $\sum_{n=0}^\infty X_n z^{-n}$ satisfying \eqref{eq:1} is a rational function with denominator $z^2 - Pz + Q$. Indeed,
2 KB (351 words) - 20:26, 20 November 2023
...tions, in particular with approximations of irrational numbers by rational numbers. Approximations of curves, surfaces, spaces and mappings are studied in geo
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...]] $\mathbf{Q}(e^{2\pi i/p})$ is not divisible by $p$. All other odd prime numbers are called irregular (see [[Irregular prime number|Irregular prime number]]
...ernoulli numbers]] $B_1,\ldots,B_{(p-3/2)}$, when these numbers (which are rational) are written as irreducible fractions (see [[#References|[a1]]]).
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which is an identity in [[formal power series]] over the rational numbers.
Over the field of $p$-adic numbers we define
1 KB (186 words) - 20:47, 23 November 2023
...for which $Q(x_1,\ldots,x_n)$ is defined. Then $Q$ is a sum of squares of rational functions with coefficients in $F$.
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$#C+1 = 101 : ~/encyclopedia/old_files/data/R077/R.0707590 Rational function
A rational function is a function $ w = R ( z) $,
8 KB (1,257 words) - 03:49, 4 March 2022
...ber field]] with a non-Abelian [[Galois group]] over the field of rational numbers $\QQ$,
of algebraic numbers, and the term "non-Abelian" is understood to refer to the Galois group ov
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...ts; moreover, any factorization of $\phi(x)$ into irreducible factors with rational coefficients leads to a factorization of $f(x)$ into irreducible factors wi
...Thus, $g(c_i)$ divides $f(c_i)$. Choosing arbitrary divisors $d_i$ of the numbers $f(c_i)$, one obtains
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The measure of algebraic independence of the numbers $\alpha_1,\dots,\alpha_m$ is the function
where the minimum is taken over all polynomials of degree at most $n$, with rational integer coefficients not all of which are zero, and of height at most $H$.
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...of degree $n$. All rational numbers, and only such numbers, are algebraic numbers of the first degree. The number $i$ is an algebraic number of the second de
...n by zero) are algebraic numbers; this means that the set of all algebraic numbers is a [[Field|field]]. A root of a polynomial with algebraic coefficients is
10 KB (1,645 words) - 17:08, 14 February 2020
...for any $x \in X \subset \mathbf{R}$ (or $x \in X \subset \mathbf{C}$) the numbers $x+T$ and $x-T$ also belong to $X$ and such that the following equality hol
The numbers $\pm nT$, where $n$ is a natural number, are also periods of $f$. For a fun
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''of algebraic numbers''
...\alpha_1,\ldots,\alpha_n$, $\beta_1,\ldots,\beta_n$ are [[Rational number|rational]] or [[algebraic number]]s and $\log\alpha_1,\ldots,\log\alpha_n$, with fix
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...cteristic zero contains a subfield isomorphic to the field of all rational numbers, and a field of finite characteristic $p$ contains a subfield isomorphic to
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...et series]] with exponents that are independent over the field of rational numbers; etc.
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When $m$ is rational, this is an [[algebraic curve]]. In particular, when $m=1$ it is a circle,
...s case the pole is a multiple point (see Fig.). When $m=p/q$ is a positive rational number, the curve consists of $p$ intersecting branches. When $m$ is a nega
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''Mahler's 3/2 problem'' concerns the existence of "Z-numbers". A ''Z-number'' is a real number $x$ such that the [[Fractional part of
...natural numbers $n$. Kurt Mahler conjectured in 1968 that there are no Z-numbers.
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...nd the rational numbers $x$ in the prime decomposition of which only prime numbers from the set $S$ appear.
...s, every element of this set is of the form $|.| v$, where $v$ is either a rational prime number or the symbol $\infty$. One now modifies the definition of the
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...cyimages/s/s085/s085000/s0850009.png" /> and if there exists a sequence of
rational integers <img align="absmiddle" border="0" src="https://www.encyclopediaofm
...pediaofmath.org/legacyimages/s/s085/s085000/s08500020.png" /> are
rational numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/
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A method for isolating the algebraic part in indefinite integrals of rational functions. Let $ P( x) $
are real numbers, $ ( p _ {j} ^ {2} /4)- q _ {j} < 0 $,
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...^G$ of $K$ consisting of all elements fixed under $G$ is itself a field of rational functions in $n$ (other) variables with coefficients in $\mathbf Q$. This q
...l, the answer to Noether's problem is negative. The first example of a non-rational field $K^G$ was constructed in [[#References|[2]]], and in this example $G$
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...a field|characteristic]] 0 is [[Isomorphism|isomorphic]] to the field of [[rational number]]s. A prime field of [[Characteristic of a field|characteristic]] $p
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...without remainder) by $b$; this is noted as $b\mid a$. Division of complex numbers is defined by the formula
while division of the complex numbers in their trigonometric form is given by the formula
3 KB (464 words) - 18:40, 30 December 2018
...is in fact $A$-rational. Recall that a formal power series $\alpha$ is $R$-rational, $R$ a commutative ring, if there exist two polynomials $P , Q \in R [ X ]$
For a rational function $R \in K ( X )$, there are several representations of the form $R
5 KB (828 words) - 11:51, 24 December 2020
...ield of [[algebraic number]]s, the [[algebraic closure]] of the field of [[rational number]]s, is an algebraic extension but not of finite degree.
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''(in the geometry of numbers)''
...tional polyhedron, i.e. is defined by a system of linear inequalities with rational coefficients, then the "non-zero volume condition" in the flatness theore
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...thmetic condition (usually one looks for solutions in integers or rational numbers). The study of such equations forms the topic of the theory of [[Diophantin
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...and it indicates the number of pairwise non-homological (over the rational numbers) cycles in it. For instance, for the sphere $S^n$:
is equal to its [[Euler characteristic|Euler characteristic]]. Betti numbers were introduced by E. Betti [[#References|[1]]].
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...o element other than the identity is (aperiodic). The additive group of [[rational number]]s $\mathbb{Q}^+$ is an aperiodic example, and the group $\mathbb{Q}
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...e construction described above gives the completion of the set of rational numbers by Dedekind sections.
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...characteristic number]] defined for closed oriented manifolds and assuming rational values. Let $ x \in H ^ {**} ( \mathop{\rm BO} ; \mathbf Q ) $
the rational number $ x [ M ] = \langle x ( \tau M ) , [ M] \rangle $
5 KB (680 words) - 08:07, 6 June 2020
...algebraic numbers (cf. [[Algebraic number|Algebraic number]]) by rational numbers: Find a quantity $\nu=\nu(n)$ such that for each algebraic number $\alpha$
has a finite number of solutions in rational integers $p$ and $q$, $q>0$, for any $\epsilon>0$, and an infinite number o
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is an integer, while each one of the numbers $ b _ {j} $,
the numbers
2 KB (331 words) - 17:32, 5 June 2020
...ree $n$ is an extension of degree $n$ of the field $\mathbf Q$ of rational numbers. Alternatively, a number field $K$ is an algebraic number field (of degree
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...c numbers and let $\alpha_1,\dots,\alpha_m$ be pairwise distinct algebraic numbers; then
...numbers, linearly independent over the field of rational numbers, then the numbers $e^{\beta_1},\dots,e^{\beta_n}$ are algebraically independent.
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...R$ of real numbers is a Euclidean field. The field $\mathbf Q$ of rational numbers is not a Euclidean field.
...D valign="top"> G.H. Hardy; E.M. Wright. "An Introduction to the Theory of Numbers", Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wile
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...ndamental sequences of rational numbers one arrives at the concept of real numbers (cf. [[Real number|Real number]]); by identifying isomorphic groups with ea
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* The field of complex numbers is quadratically closed; more generally, any [[algebraically closed field]]
* The field of real numbers is not quadratically closed as it does not contain a square root of $-1$.
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...of the concept of irrationality (cf. [[Irrational number]]). Thus, the two numbers $\alpha$ and $1$ are linearly independent if and only if $\alpha$ is irrati
2 KB (325 words) - 19:52, 20 November 2014
...ing $\mathbf Z$ is the minimal ring which extends the semi-ring of natural numbers $\mathbf N=\{1,2,\dots\}$, cf. [[Natural number|Natural number]]. Cf. [[Num
...s an algebraic field extension, where $\mathbf Q$ is the field of rational numbers, the [[field of fractions]] of $\mathbf Z$, then the integers of $k$ are th
2 KB (283 words) - 17:19, 30 November 2014
...field]] $\mathbf Q(e^{2\pi i/p})$ is divisible by $p$. All other odd prime numbers are called regular.
...one of the numerators of the first $(p-3)/2$ [[Bernoulli numbers|Bernoulli numbers]] $B_2,B_4,\dots,B_{p-3}$ is divisible by $p$ (cf. [[#References|[1]]]).
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An Abelian [[Extension of a field|extension]] of the field of rational numbers $\mathbf{Q}$, i.e. a [[Galois extension]] $K$ of $\mathbf{Q}$ such that the
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...an infinite number of solutions in integers $q \ge 1$ for almost-all real numbers $\alpha$ if the series
...s. For example, for almost-all $\alpha$ there exists an infinite number of rational approximations $p/q$ satisfying the inequality
8 KB (1,172 words) - 17:12, 8 March 2018
variables with integer rational coefficients not all of which are divisible by $ m $.
are different prime numbers, is equivalent to the solvability of the congruences
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...sponds a unique real logarithm (logarithms of negative numbers are complex numbers). The main properties of the logarithm are:
These make it possible to reduce multiplication and division of numbers to the addition and subtraction of their logarithms, and the raising to pow
3 KB (453 words) - 15:11, 19 August 2014
over an algebraically closed field is a [[Rational curve|rational curve]], i.e. it is birationally isomorphic to the projective line $ P ^
the rational mapping $ \phi _ {| K _ {X} | } : X \rightarrow P ^ {g-1} $
3 KB (483 words) - 06:28, 31 March 2023
...\alpha_n$ be arbitrary real numbers and let $N$ and $\epsilon$ be positive numbers; then there are integers $m$ and $p_1,\ldots,p_n$ such that
2 KB (349 words) - 12:10, 13 March 2018
==Algebraic independence of numbers.==
Complex numbers $ \alpha _{1} \dots \alpha _{n} $
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...own. A classical example of such a sequence is the sequence of [[Fibonacci numbers]] $1,1,2,3,5,8$ defined by $a_{n+2}=a_{n+1}+a_n$ with $a_0=0$, $a_1=1$.
...orm a recursive sequence. Such a series represents an everywhere-defined [[rational function]]: its denominator is the reciprocal polynomial $X^p F(1/X)$.
2 KB (283 words) - 16:38, 30 December 2018
...om]] for the real line can be formulated in terms of Dedekind cuts of real numbers.
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for different odd prime numbers $p$ and $q$. There are two additions to this quadratic reciprocity law, nam
...laws in quadratic extensions $\mathbf Q(\sqrt d)$ of the field of rational numbers, since the factorization into prime factors in $\mathbf Q(\sqrt d)$ of a pr
2 KB (304 words) - 19:26, 14 August 2014
...]; 3) $a_{n+2}=a_{n+1}+a_n$, the sequence of [[Fibonacci numbers|Fibonacci numbers]].
...only if the formal power series $\alpha(x)=\alpha_0+\alpha_1x+\dotsb$ is a rational function of the form $\alpha(x)=p(x)/q(x)$, with $p(x)=1-p_1x-\dotsb-p_mx^m
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...he same kind, and which is accepted as a unit" . Rigorous theories of real numbers were constructed at the end of the 19th century by K. Weierstrass, G. Canto
Real numbers form a non-empty totality of elements which contains more than one element
26 KB (4,086 words) - 09:51, 4 April 2020
...Cantor's construction of the set of real numbers from the set of rational numbers.
2 KB (257 words) - 17:46, 4 June 2020
...ional variety|Unirational variety]]). Since Abelian varieties can never be rational, the main interest is in rationality theorems for linear algebraic groups.
of complex numbers were in fact proved by E. Picard and, in contemporary terminology, establis
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...ample, any real number is an accumulation point of the set of all rational numbers in the ordinary topology. In a [[discrete space]], no set has an accumulati
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geometry, which reduces the problem of the existence of rational
non-trivial absolute valuations $\nu$ on $K$ the set of $K_\nu$-rational
3 KB (543 words) - 15:48, 17 February 2012
...ctively defined. Thus, in the case of rational approximations to algebraic numbers, the bound for the denominators of "good" approximations, which is establ
...olves the application of effective methods of the theory of transcendental numbers (cf. [[Linear form in logarithms|Linear form in logarithms]]). The best res
6 KB (940 words) - 18:12, 23 November 2014
are finite or infinite sequences of complex numbers or functions. For continued fractions one uses the notation
...w.encyclopediaofmath.org/legacyimages/c/c025/c025640/c02564026.png" /> are rational functions. Famous examples of explicit continued fractions are those for hy
6 KB (886 words) - 17:03, 7 February 2011
...hematical novelette [[#References|[a7]]]. Only in later years have surreal numbers become the subject of more traditional mathematical papers and books [[#Ref
...nes. In fact, the construction never terminates, and therefore the surreal numbers do not form a set but a proper class (cf. [[Types, theory of|Types, theory
8 KB (1,226 words) - 16:11, 3 July 2016
...{ n } ( z , \tau )$, and as weights the [[Christoffel numbers|Christoffel numbers]]
...ces|[a2]]]. The underlying ideas have been generalized from polynomials to rational functions. See [[#References|[a1]]].
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...[6]</TD> <TD valign="top"> A.O. Gel'fond, "Transcendental and algebraic numbers" , Dover, reprint (1960) (Translated from Russian)</TD></TR></table>
...theorem to the problem of simultaneous approximation of several algebraic numbers. This was extended by H.P. Schlickewei [[#References|[a2]]] to include $p$-
4 KB (557 words) - 18:10, 23 November 2014
...to a [[Field|field]], for example, the field of rational, real or complex numbers.
...into factors of the first and second degree, and over the field of complex numbers into factors of the first degree (cf. [[Algebra, fundamental theorem of|Alg
9 KB (1,497 words) - 10:44, 27 June 2015
...additive number theory. Let $X_1,\ldots,X_k$ be arbitrary sets of natural numbers, let $N$ be a natural number and let $J_k(N)$ be the number of solutions of
where $n_1\in X_1,\ldots,n_k\in X_k$. It is with the investigation of the numbers $J_k(N)$ that additive number theory is concerned; for example, if it can b
5 KB (868 words) - 21:24, 18 November 2016
has rational coefficients,
is called a rational trigonometric sum; if $ P = q $,
4 KB (621 words) - 08:26, 6 June 2020
...] in connection with the calculation of the sum of equal powers of natural numbers:
The values of the first Bernoulli numbers are:
4 KB (684 words) - 18:44, 5 October 2023
The Apéry numbers $a _ { n }$, $b _ { n }$ are defined by the finite sums
...irrationality proof of $\zeta ( 3 )$, motivated by the shape of the Apéry numbers. Despite much efforts by many people there is no generalization to an irrat
3 KB (525 words) - 20:51, 23 January 2024
...s of primes: $\theta(x)$ is the sum of the natural logarithms of the prime numbers up to $x$, while $\psi(x)=\sum_{n\leq x}\Lambda(n)$ (cf. [[Chebyshev functi
Arithmetic functions appear and are employed in studies on the properties of numbers. However, the theory of arithmetic functions is also of independent interes
4 KB (608 words) - 08:18, 4 November 2023
A conjecture on the finiteness of the set of rational points on an [[Algebraic curve|algebraic curve]] of genus $ g > 1 $.
...'s conjecture is taken to be the assertion of the finiteness of the set of rational points $ X ( L) $
7 KB (1,068 words) - 08:01, 6 June 2020
...\lim_{r_n\to\a} a^{r_n}$, where $r_n$ is an arbitrary sequence of rational numbers tending to $\a$. Powers with a complex base (see
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...p^\nu})$ is the field obtained by adjoining to the field $\mathbb{Q}$ of [[rational number]]s a primitive root of unity $\zeta_{p^\nu}$ of order $p^\nu$ and $k
2 KB (327 words) - 11:17, 9 April 2023
group of the field of rational numbers are sequences of the form
The multiplicative group of the field of rational numbers is
4 KB (757 words) - 21:21, 22 November 2014
If $k$ is the field of rational numbers, the problem becomes one of constructing a normal algebraic number field wi
...bf C(X)$, the field of rational functions in one variable over the complex numbers, which is well under control. Under certain conditions on the group one can
4 KB (586 words) - 04:07, 25 February 2022
...ortant examples of divisible groups are the additive group of all rational numbers and the group of all complex roots of unity of degrees $p^k$, $k=1,2,\ldots
...ive module|Injective module]]). Let $\mathbf Q_p$ be the field of $p$-adic numbers and $\mathbf Z_p$ its ring of integers. Then the quasi-cyclic group for the
2 KB (335 words) - 17:07, 30 July 2014
...equations|Diophantine equations]] arises from the theory of transcendental numbers, from the solution by A.O. Gel'fond of Hilbert's seventh problem and from s
the empty set. For instance, for the field of rational numbers, given a finite set $ S = \{ p _ {1} \dots p _ {s} \} $
7 KB (1,087 words) - 19:41, 5 June 2020
is the greatest common divisor of the numbers $ n $
It follows immediately from Euler's criterion that the numbers $ 1 \dots p - 1 $
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==Liouville's theorem on approximation of algebraic numbers==
...r]] of degree $n \ge 2$ and $p$ and $q$ are any positive integral rational numbers, then
8 KB (1,240 words) - 04:55, 24 February 2022
...ree operations (addition, multiplication, passing to the limit by rational numbers), performed not more than a countable number of times, starting from an arg
1 KB (210 words) - 13:18, 12 December 2013
...set of Lagrange constants in the problem of rational approximation to real numbers. The Lagrange spectrum is strictly contained in the Markov spectrum (see [[
1 KB (191 words) - 16:51, 23 November 2023
of a recurrence is a rational function $ { {r ( X ) } / {s ( X ) } } $
If so, the distinct complex numbers $ \alpha _ {i} $
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rational numbers $\Q$, the field of real numbers $\R$, the field of
complex numbers $\C$, finite fields (see
6 KB (929 words) - 00:28, 18 May 2013
...ho defined the space of real numbers as the completion of that of rational numbers, see [[Real number|real number]]
2 KB (340 words) - 15:24, 18 October 2014
...n|Simply-connected domain]]) is called rational if its homotopy groups are rational vector spaces (cf. also
...e of homotopy categories between the homotopy category of simply-connected rational spaces and the homotopy category of connected differential graded Lie algeb
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