"무리수와 초월수"의 두 판 사이의 차이

2009년 6월 16일 (화) 16:19 판

간단한 소개

먼저 대수

린데만-바이어슈트라스 정리

린데만-바이어슈트라스 정리

겔퐁드-슈나이더 정리

If α and β are algebraic numbers (with α≠0 and \(\log \alpha\) any non-zero logarithm of α), and if β is not a rational number, then any value of \(\alpha^{\beta} = \exp\{\beta \log \alpha\}\) is a transcendental number. ===Comments=== * The values of \(\alpha\) and \(\beta\) are not restricted to real numbers; all complex numbers are allowed. * In general, \(\alpha^{\beta} = \exp\{\beta \log \alpha\}\) is multivalued, where "log" stands for the complex logarithm. This accounts for the phrase "any value of" in the theorem's statement. * An equivalent formulation of the theorem is the following: if \(\alpha\) and \(\gamma\) are nonzero algebraic numbers, and we take any non-zero logarithm of α, then \((\log \gamma)/(\log \alpha)\) is either rational or transcendental. * If the restriction that \(\beta\) be algebraic is removed, the statement does not remain true in general (choose \(\alpha=3\) and \(\beta=\log 2/\log 3\), which is transcendental, then \(\alpha^{\beta}=2\) is algebraic). A characterization of the values for α and β which yield a transcendental α^β is not known.

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-<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">간단한 소개</h5>
+<h5>간단한 소개</h5>
+먼저 대수
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 <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">린데만-바이어슈트라스 정리</h5>
-대수적 수 <math>\alpha_1,\cdots,\alpha_n</math> 가 유리수체 위에서 일차독립이면, <math>e^{\alpha_1},\cdots,e^{\alpha_n}</math> 는 유리수체 위에서 대수적으로 독립이다.
+* [[린데만-바이어슈트라스 정리]]<br>
-(또는 서로 다른 대수적수 <math>\alpha_1,\cdots,\alpha_n</math> 에 대하여, <math>e^{\alpha_1},\cdots,e^{\alpha_n}</math> 는 대수적수체 위에서 일차독립이다.)
-<h5><br> 겔퐁드-슈나이더 정리</h5>
+<h5>겔퐁드-슈나이더 정리</h5>
+:If α and β are [[algebraic number]]s (with α≠0 and <math>\log \alpha</math> any non-zero logarithm of α), and if β is not a [[rational number]], then any value of <math>\alpha^{\beta} = \exp\{\beta \log \alpha\}</math> is a [[transcendental number]]. ===Comments=== * The values of <math>\alpha</math> and <math>\beta</math> are not restricted to [[real number]]s; all [[complex number]]s are allowed. * In general, <math>\alpha^{\beta} = \exp\{\beta \log \alpha\}</math> is [[multivalued function|multivalued]], where "log" stands for the [[complex logarithm]]. This accounts for the phrase "any value of" in the theorem's statement. * An equivalent formulation of the theorem is the following: if <math>\alpha</math> and <math>\gamma</math> are nonzero algebraic numbers, and we take any non-zero logarithm of α, then <math>(\log \gamma)/(\log \alpha)</math> is either rational or transcendental. * If the restriction that <math>\beta</math> be algebraic is removed, the statement does not remain true in general (choose <math>\alpha=3</math> and <math>\beta=\log 2/\log 3</math>, which is transcendental, then <math>\alpha^{\beta}=2</math> is algebraic). A characterization of the values for α and β which yield a transcendental α<sup>β</sup> is not known.
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 * [[1964250|0 토픽용템플릿]]<br>
 ** [[2060652|0 상위주제템플릿]]<br>
-**   <br>
+** [[린데만-바이어슈트라스 정리]]<br>
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 <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">동영상</h5>
-* http://www.youtube.com/results?search_type=&search_query=
+* http://www.youtube.com/results?search_type=&search_query=<br>
-* <br>