...]] $\mathbf{Q}(e^{2\pi i/p})$ 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
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...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
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...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 $
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...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
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...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
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...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
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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
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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} $
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...\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
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==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 $X^p F(1/X)$.
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...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
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...]; 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
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