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Revision difference : algebraic numbers are countable |
| Version 8 |
Version 7 |
| \begin{thmplain} |
\begin{thmplain} |
| The set of (a) all algebraic numbers, (b) the real algebraic numbers is countable. |
The set of (a) all algebraic numbers, (b) the real algebraic numbers is countable. |
| \end{thmplain} |
\end{thmplain} |
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| {\em Proof.} \,Let's consider the algebraic equations |
{\em Proof.} \,Let's consider the algebraic equations |
| \begin{align} |
\begin{align} |
| P(x) = 0 |
P(x) = 0 |
| \end{align} |
\end{align} |
| where |
where |
| $$P(x) := a_0x^n+a_1x^{n-1}+...+a_{n-1}x+a_n$$ |
$$P(x) := a_0x^n+a_1x^{n-1}+...+a_{n-1}x+a_n$$ |
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is an \PMlinkname{irreducible}{IrreduciblePolynomial2} and primitive polynomial with integer coefficients $a_j$ and
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is an \PMlinkname{irreducible}{IrreduciblePolynomial} and primitive polynomial with integer coefficients $a_j$ and
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| \,$a_0 > 0$. \,Each algebraic number \PMlinkescapetext{satisfies} exactly one such equation (see the minimal polynomial). \,For every integer \,$N = 2,\,3,\,4,\,...$\, there exists a finite number of equations (1) such that |
\,$a_0 > 0$. \,Each algebraic number \PMlinkescapetext{satisfies} exactly one such equation (see the minimal polynomial). \,For every integer \,$N = 2,\,3,\,4,\,...$\, there exists a finite number of equations (1) such that |
| $$n+a_0+|a_1|+...+|a_n| = N$$ |
$$n+a_0+|a_1|+...+|a_n| = N$$ |
| (e.g. if \,$N = 3$, then one has the equations \,$x-1 = 0$\, and \,$x+1 = 0$) and thus only a finite set of algebraic numbers as the \PMlinkescapetext{roots} of these equations. \,These algebraic numbers may be ordered to a \PMlinkname{finite sequence}{OrderedTuplet} $S_N$ using a \PMlinkescapetext{fixed ordering} system, for example by the magnitude of the real part and the imaginary part. \,When one forms the concatenated sequence |
(e.g. if \,$N = 3$, then one has the equations \,$x-1 = 0$\, and \,$x+1 = 0$) and thus only a finite set of algebraic numbers as the \PMlinkescapetext{roots} of these equations. \,These algebraic numbers may be ordered to a \PMlinkname{finite sequence}{OrderedTuplet} $S_N$ using a \PMlinkescapetext{fixed ordering} system, for example by the magnitude of the real part and the imaginary part. \,When one forms the concatenated sequence |
| $$S_2,\,S_3,\,S_4,\,...$$ |
$$S_2,\,S_3,\,S_4,\,...$$ |
| it comprises all algebraic numbers in a countable setting, which defines a bijection from the set onto $\mathbb{Z}_+$. |
it comprises all algebraic numbers in a countable setting, which defines a bijection from the set onto $\mathbb{Z}_+$. |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{EK}{\sc E. Kamke:} {\em Mengenlehre}. \,Sammlung G\"oschen band 999/999a. \,-- Walter de Gruyter \& Co., Berlin (1962). |
\bibitem{EK}{\sc E. Kamke:} {\em Mengenlehre}. \,Sammlung G\"oschen band 999/999a. \,-- Walter de Gruyter \& Co., Berlin (1962). |
| \end{thebibliography} |
\end{thebibliography} |
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