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¤@­Ó½Æ¼Æ x ¥s°µ¶W¶V¼Æ(transcendental number) ªº·N«ä¬O»¡¡A¹ï©ó¥ô¦ó Q ¤Wªº¦h¶µ¦¡ f(T)¡A­Y f(x)=0¡A«h $f(T)\equiv 0$¡C§_«h¡A´N¥s°µ¥N¼Æ¼Æ(algebraic numnbers)¡C

¤£Ãø¬Ý¥X¡Ax ¬O¥N¼Æ¼Æªº¥R¤À¥²­n±ø¥ó¬O $[\mathbf{Q}(x):\mathbf{Q}]<\infty$¡C ¦]¬°¡A­Y [Q(x):Q]=n¡A«h 1,x,¡K,xn+1¡A¤£¥i¯à¦b Q ¤W½u©Ê¿W¥ß¡C ©Ò¥H¥i¥H§ä¨ì¤£¥þ¬°¹sªº¦³²z¼Æ $\alpha_0$,$\alpha_1$,¡K,$\alpha_n$¡A¨Ï±o $\alpha_0x^{n+1}+\alpha_1x^n+\cdots+\alpha_nx+\alpha_{n+1}=0$¡C©Ò¥H x ¬O¥N¼Æ¼Æ¡C

©Ò¦³¥N¼Æ¼Æ§Î¦¨ªº¶°¦X¡AÅܦ¨¤@­ÓÅé¡C¡]ÃÒ©ú¤£¬O«Ü®e©ö¡A¦ý¬OŪªÌ­È±o¦Û¤v°µ°µ¬Ý¡C¡^ 5

¦pªG x ¬O¶W¶V¼Æ¡A«h Q(x) ªº¥|«h¹Bºâ©M¦³²z¨ç¼Æ

\begin{displaymath}
\mathbf{Q}(T) = \{ \frac{f(T)}{g(T)}:f(T), g(T) \mbox{{\font...
...0.1pt{\fontfamily{cwM1}\fontseries{m}\selectfont \char 31}} \}
\end{displaymath}

ªº¥|«h¹Bºâ§¹¥þ¤@¼Ë¡A¨Ò¦p (1+x)(1-x)=1-x2¡C¦ý¬O¦pªG x ¬O¥N¼Æ¼Æ¡A¨Ò¦p $x=\sqrt{2}$¡A«h

\begin{eqnarray*}
(1+\sqrt{2})(1-\sqrt{2})&=&1-(\sqrt{2})^2\\
&=&1-2\\
&=&-1
\end{eqnarray*}


§Ú­Ì¥i¥H§â $x^2=(\sqrt{2})^2$ Ä~Äò¤Æ²¡C

¶W¶V¼Æ©MµL²z¼Æ¨Ã¤£¤@¼Ë¡C¦³²z¼Æ¬O¥i¥H¼g¦¨ $\frac{q}{p}$ «¬¦¡ªº¼Æ¡A¨ä¤¤ p »P q ¬O¾ã¼Æ¡A$p\neq 0$¡C

\begin{displaymath}\begin{eqalign}
\mbox{{\fontfamily{cwM0}\fontseries{m}\select...
...tfamily{cwM1}\fontseries{m}\selectfont \char 98}}
\end{eqalign}\end{displaymath}

¦p $\sqrt[3]{2}$ ¬O¥N¼Æ¼Æ¡A$\pi i$ ¬O¶W¶V¼Æ¡Ai ¬O¥N¼Æ¼Æ¦Ó¤£¬O¹ê¼Æ¡C

ªk°ê¼Æ¾Ç®a Joseph Liouville¡]1809¡ã1882¦~¡^¦b1844¦~§ä¥X²Ä¤@­Ó¶W¶V¼Æ¥X¨Ó¡CLiouville ªº¤èªk¬O®Ú¾Ú¤@­Ó°ò¥»ªº·§©À¡G¥N¼Æ¼Æ¬OµLªk¥Î¦³²z¼Æ¨Ó°ª«×¹Gªñªº¡C§óºë½Tªº»¡¡A¥LÃÒ©ú

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­Y¹ê¼Æ £i ¬O¤@­Óº¡¨¬

\begin{displaymath}
a_0\xi^n+a_1\xi^{n-1}+\cdots+a_{n-1}\xi+a_n=0
\end{displaymath}

ªº¥N¼Æ¼Æ¡Aai ¬O¾ã¼Æ¡A$a_0\neq 0$¡C«h¥²¦s¦b¤@­Ó¥¿¼Æ K¡A¨Ï±o $\vert\xi-\frac{q}{p}\vert>\frac{K}{p^n}$¡A ¨ä¤¤ $\frac{q}{p}$ ¬O¨¬°÷¾aªñ £i ªº¦³²z¼Æ¡A¥B $\frac{q}{p}\neq\xi$¡C

ÃÒ©ú
³] f'(T) ¬O f(T) ªº¾É¨ç¼Æ¡A¨ä¤¤

\begin{displaymath}
f(T)=a_0T^n+a_1T^{n-1}+\cdots+a_{n-1}T+a_n \; .
\end{displaymath}

§ä¤@­Ó¥¿¼Æ M¡A¨Ï±o¥u­n $\xi-1<u<\xi+1$¡A´N·|¦³ $\vert f'(u)\vert<\frac{1}{M}$¡C

­Y $\frac{q}{p}$ ¨¬°÷¾aªñ £i¡A¨Ï±o

\begin{displaymath}
\xi-1<\frac{q}{p}<\xi+1 \; \mbox{{\fontfamily{cwM0}\fontseries{m}\selectfont \char 47}} \; f(\frac{q}{p})\neq 0
\end{displaymath}

«h

\begin{displaymath}
\vert f(\frac{q}{p})\vert=\frac{\vert a_0q^n+a_1q^{n-1}p+\cdots+a_{n-1}qp^{n-1}+a_np^n\vert}{p^n}\geq\frac{1}{p^n}
\end{displaymath}

¤S

\begin{displaymath}
f(\frac{q}{p})=f(\frac{q}{p})-f(\xi)=(\frac{q}{p}-\xi)\cdot f'(\xi)
\end{displaymath}

¨ä¤¤ $\xi-1<\xi_1<\xi+1$¡A¬G

\begin{displaymath}
\vert\xi-\frac{q}{p}\vert = \frac{\vert f(\frac{q}{p})\vert}{f'(\xi_1)} > \frac{M}{p^n}
\end{displaymath}

±oÃÒ¡C

§Q¥Î¥H¤W©w²z¡ALiouville «ü¥X

\begin{displaymath}
\xi=0.110001000\cdots=\frac{1}{10}+\frac{1}{10^{2!}}+\frac{1}{10^{3!}}+\cdots+\frac{1}{10^{n!}}+\cdots
\end{displaymath}

¬O¤@­Ó¶W¶V¼Æ¡C¨Ò¦p¡A°²³] £i º¡¨¬¤@­Ó57¦¸ªº¤èµ{¼Æ¡A¥B K=1000¡A¨º»ò§Ú­Ì¥u­n¨ú¤@­Ó¦³²z¼Æ £b¡A£b ¬O £i ªº«e57¶µªº©M¡A§Y

\begin{displaymath}
\eta=\frac{1}{10}+\frac{1}{10^{2!}}+\frac{1}{10^{3!}}+\cdots+\frac{1}{10^{57!}}=\frac{q}{10^{57!}}
\end{displaymath}

q ¬O¬Y­Ó¾ã¼Æ¡C «h

\begin{displaymath}
\vert\xi-\eta\vert=\frac{1}{10^{58!}}+\frac{1}{10^{59!}}+\cdots<2\frac{1}{10^{58!}}<\frac{1000}{(10^{57!})^{57}}
\end{displaymath}

¦]¦¹ Liouville ©w²z¤£¦¨¥ß¡C¬G £i ¤£±o¤£¬O¶W¶V¼Æ¡C

¤T¤Q¦~¥H«á¡AGeorg Cantor §Q¥Î¶°¦X½×ªº¤èªkÃÒ©ú¡A©Ò¦³ªº¹ê¼Æªº¥N¼Æ¼Æ¬O¥i¼Æªº (countable) ¦Ó¹ê¼Æ¬O¤£¥i¼Æªº (uncountable)¡A¦]¦¹¥²¦³¤@­Ó¶W¶V¼Æ¦s¦b¡C

1873¦~ªk°ê¼Æ¾Ç®a Charles Hermite¡]1822¡ã1901¦~¡^ÃÒ©ú

\begin{displaymath}
e \quad ( = \lim_{n\rightarrow\infty}(1+\frac{1}{n})^n)
\end{displaymath}

¬O¶W¶V¼Æ¡C

1882¦~ F. Lindemann §Q¥Î Hermite ªº·Qªk¡AÃÒ©ú £k ¬O¶W¶V¼Æ¡C¨Æ¹ê¤W Lindemann ÃÒ©ú¡G ­Y $a\neq 0$ ¬O¤@­Ó¥N¼Æ¼Æ¡A«h ea ¬O¤@­Ó¶W¶V¼Æ¡C¨ú a=1¡A±oÃÒ e ¬O¶W¶V¼Æ¡C ¨ú $a=i\pi$¡A«h $e^{i\pi}=\cos\pi+i\sin\pi=1$ ¤£¬O¶W¶V¼Æ¡A¬G $i\pi$ ¤£¬O¥N¼Æ¼Æ¡F±oÃÒ £k ¬O¶W¶V¼Æ¡C

1900¦~ David Hilbert¡]1862¡ã1943¦~¡^¦b¤Ú¾¤ªº°ê»Ú¼Æ¾Ç·|´£¥X¦³¦Wªº Hilbert ªº23­Ó°ÝÃD¡C ¥Lªº²Ä¤C­Ó°ÝÃD¬O¡G¦pªG a »P b ³£¬O¥N¼Æ¼Æ¡A¥B $a\neq 0,1$¡Ab ¬OµL²z¼Æ¡A«h ab ¬O§_¬°¶W¶V¼Æ¡H¨Ò¦p¡A $2^{\sqrt{2}}$¡B $e^{-\pi}=e^{i\pi\cdot i}= (-1)^i$¡B $e^{i\pi b}=(-1)^b$ ¬O¤£¬O¶W¶V¼Æ¡H

¦b1934¦~¡A«X°ê¼Æ¾Ç®a A.O. Gelfond »P¼w°ê¼Æ¾Ç®a T. Schneider ¤À§O¸Ñ¨M¤F Hilbert ²Ä¤C°ÝÃD¡Aµª®×¬OªÖ©wªº¡C

­nÃÒ©ú¬Y¨Ç¼Æ¬O¶W¶V¼Æ¨Ã¤£¬O«Ü®e©öªº¡C¨Ò¦p¡A0.123456789101112¡K ¬O¤£¬O¶W¶V¼Æ¡H¡]µª®×¡G¬O¡C¡^ ¤S¦p¡G$e+\pi$¡Bee¡B$\pi^{\pi}$¡B$\pi^e$¡B $2^{2^{\sqrt{2}}}$¡B$2^{\pi}$¡B2e¡BEuler ±`¼Æ¡]= $\lim_{n\rightarrow\infty} (1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n)$¡^¬O¤£¬O¶W¶V¼Æ¡H¨Æ¹ê¤W¡A¨ì²{¦b¬°¤î¡A§Ú­Ì³s³o¨Ç¼Æ¬O¤£¬OµL²z¼Æ³£¤£ª¾¹D¡C

¥Ñ©ó¶W¶V¼Æªº¬ã¨s¡A¼Æ¾Ç®a³s±aªº¬ã¨s¡A¥N¼Æ¼Æªº¹ï¼Æªº½u©Ê²Õ¦X¦ô­p­È¡A¾ã«Y¼Æ¦h³»¦¡¸Ñªººë½T½d³ò¡A¦³²z¼Æ¹Gªñ¥N¼Æ¼Æªº¤è¦¡¡C¦P®É¼Æ¾Ç®a¤]³]­p¥X³\¦h¥©§®ªº¤èªk¨Ó°Q½×¶W¶V¼Æ¡CHilbert ¦b´£¥X²Ä¤C°ÝÃD®É¯S§O±j½Õ¡A­n¸Ñ¨M³o­Ó°ÝÃD¶Õ¥²µo²{³\¦h·sªº¤èªk»P·sªº¨¤«×¨ÓÁA¸ÑµL²z¼Æ»P¶W¶V¼Æªº¥»½è¡C

¦³Ãö Lindemann ©w²zªºÃÒ©ú¡AŪªÌ¥i°Ñ¦Ò¥H¤U¸ê®Æ¡A

S. Lang,¡mAlgebra¡n, Appendix, 492¡ã499­¶¡A³oùتºÃÒ©ú¬O±Ä¥Î Gelfond »P Schneidel ¤èªk¡CŪªÌ¥u­n¨ã³Æ½ÆÅܨç¼Æªºª¾ÃÑ´N¥i¬ÝÀ´¡C¨Ã¥B¡A´N¦b³oùØ¡ASchanuel ²q´ú²Ä¤@¦¸¥X²{¦b²³¤H²´«e¡ASchanuel ²q´ú´X¥G²ÎÄá¶W¶V¼Æ²z½×¤¤³\¦h¥D­nªº©w²z¡A¦p Lindemann n ¤¸¯À©w²z¡BGelfond-Schneider ©w²z¡BBaker n ¤¸¯À©w²z¡C¥i±¤³o­Ó²q´ú¨ì¤µ¤ÑÁÙ¨S¦³ÃÒ©ú¥X¨Ó¡C

   

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