...]] $\mathbf{Q}(e^{2\pi i/p})$ 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
3 KB (574 words) - 18:14, 14 June 2023
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 $G$
4 KB (603 words) - 17:59, 23 November 2014
...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