...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.
1 KB (190 words) - 14:18, 12 November 2023
''(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
1 KB (242 words) - 21:16, 8 April 2018
...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
608 bytes (91 words) - 17:19, 7 February 2011
...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]]].
1 KB (172 words) - 13:05, 14 February 2020
...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}
667 bytes (99 words) - 20:32, 18 November 2023
...e construction described above gives the completion of the set of rational numbers by Dedekind sections.
2 KB (347 words) - 14:30, 18 October 2014
...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
4 KB (634 words) - 15:17, 14 February 2020
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.
3 KB (379 words) - 15:19, 19 August 2014
...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
2 KB (243 words) - 18:11, 14 October 2023
...ndamental sequences of rational numbers one arrives at the concept of real numbers (cf. [[Real number|Real number]]); by identifying isomorphic groups with ea
2 KB (216 words) - 17:08, 7 February 2011
* 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 $-1$.
3 KB (439 words) - 16:55, 25 November 2023
...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