...]]s, the [[algebraic closure]] of the field of [[rational number]]s, is an algebraic extension but not of finite degree.
...only if $M/K$ is algebraic; (ii) $M / K,\,L/K $ algebraic implies $ ML/L$ algebraic.
1 KB (190 words) - 14:18, 12 November 2023
...very number field contains infinitely many elements. The field of rational numbers is contained in every number field.
...omplex numbers, or Gaussian numbers (cf. [[Gauss number]]). The set of all numbers of the form $H(\alpha)/F(\alpha)$, $F(\alpha)\neq0$, forms a number field,
2 KB (261 words) - 20:42, 23 November 2023
...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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...The algebraic closure of the field of real numbers is the field of complex numbers (cf. [[Algebra, fundamental theorem of|Algebra, fundamental theorem of]]).
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An anisotropic algebraic group over a [[Field|field]] $k$ is a
[[Linear algebraic group|linear algebraic group]] $G$ defined over $k$
1 KB (217 words) - 00:03, 24 December 2011
...f degree at most $n$) over $\mathbf Q$. (Cf. also [[Algebraic number]]; [[Algebraic number theory]]; [[Extension of a field]]; [[Number field]].)
<TR><TD valign="top">[1]</TD> <TD valign="top"> E. Weiss, "Algebraic number theory" , McGraw-Hill (1963) pp. Sects. 4–9</TD></TR>
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...nscendental number]]) at algebraic points $z\neq0$ (cf. [[Algebraic number|Algebraic number]]); proved by F. Lindemann in 1882. The following more general asser
...c numbers and let $\alpha_1,\dots,\alpha_m$ be pairwise distinct algebraic numbers; then
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...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
...rational fractions. More precisely, for any irrational [[Algebraic number|algebraic number]] $\xi$ of degree $n$ there exists a $c>0$ such that for any integer
2 KB (331 words) - 10:10, 13 April 2014
...a finite extension of the field $\mathbb{Q}$ of [[rational number]]s (an [[algebraic number field]]).
...p">[1]</TD> <TD valign="top"> J.W.S. Cassels (ed.) A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1986)</TD></TR>
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...an element $b$ such that $ab=ba=1$. In the theory of algebraic numbers and algebraic functions such elements are also called units.
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...ity of a sequence is a measure of what part of the sequence of all natural numbers belongs to a given sequence $A=\{a_k\}$ of integers $a_0=0<1\leq a_1<\dotsb
...nces $A=\{a_k\}$ and $B=\{b_t\}$, i.e. the set $A+B=\{a_k+b_t\}$ where the numbers $a_k+b_t$ are taken without repetition. If $A=B$, one puts $2A=A+A$, and si
3 KB (461 words) - 11:41, 14 February 2020
...mple, 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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...unds on these parameters. For a transcendental number $\omega$ and natural numbers $n$ and $H$, the measure of transcendency is
...l numbers $\omega$ (see [[Mahler problem|Mahler problem]]). Transcendental numbers can be classified on the basis of the difference in asymptotic behaviour of
2 KB (342 words) - 12:47, 20 December 2014
if it has no proper algebraic extension (cf.
there exists a unique (up to isomorphism) algebraic extension of $k$
1 KB (201 words) - 21:31, 5 March 2012
...a group; the corresponding maps in a [[monoid]], unital [[ring]] and other algebraic structures: see [[Conjugate elements]].
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...hod for investigating the arithmetical properties of the values assumed at
algebraic points by [[E-function|$E$-function]]s that satisfy linear differential equ
...cyclopediaofmath.org/legacyimages/s/s085/s085000/s08500011.png" /> is an [[
algebraic integer]] for <img align="absmiddle" border="0" src="https://www.encycloped
16 KB (2,130 words) - 07:52, 11 December 2016
...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
...pendent; otherwise it is called algebraically dependent. The definition of algebraic independence may be extended to the case where $ K $
==Algebraic independence of 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
2 KB (344 words) - 18:24, 30 December 2018
A one-dimensional [[Algebraic variety|algebraic variety]], defined over an algebraically closed field $k$, whose field of r
When $k$ is the field $\mathbf C$ of complex numbers, the (only) non-singular complete rational curve $X$ is the Riemann sphere
1 KB (191 words) - 10:10, 2 November 2014