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algebraic numbers are countable
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(Theorem)
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Proof. Let's consider the algebraic equations
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(1) |
where $$P(x) \;:=\; a_0x^n\!+\!a_1x^{n-1}\!+\!\ldots\!+\!a_{n-1}x\!+\!a_n$$ is an irreducible and primitive polynomial with integer coefficients $a_j$ and $a_0 > 0$ . Each algebraic number satisfies exactly one such equation (see the minimal polynomial). For every integer $N =
2,\,3,\,4,\,\ldots$ there exists a finite number of equations (1) such that $$n\!+\!a_0\!+\!|a_1|\!+\ldots+\!|a_n| \;=\; N$$ (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 roots of these equations. These algebraic numbers may be ordered to a finite sequence $S_N$ using a fixed ordering system, for example by the magnitude of the real part and the imaginary part. When one forms the concatenated sequence $$S_2,\,S_3,\,S_4,\,\ldots$$ it comprises all algebraic numbers in a countable setting, which defines a bijection from the set onto $\mathbb{Z}_+$ .
- 1
- E. KAMKE: Mengenlehre. Sammlung Göschen: Band 999/999a. - Walter de Gruyter & Co., Berlin (1962).
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"algebraic numbers are countable" is owned by pahio.
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Cross-references: onto, bijection, sequence, imaginary part, real part, finite set, number, finite, minimal polynomial, equation, coefficients, integer, primitive polynomial, algebraic equations, proof, countable, real, algebraic numbers
There are 4 references to this entry.
This is version 11 of algebraic numbers are countable, born on 2005-05-02, modified 2009-09-30.
Object id is 6999, canonical name is AlgebraicNumbersAreCountable.
Accessed 13667 times total.
Classification:
| AMS MSC: | 03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers) | | | 11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers) |
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Pending Errata and Addenda
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