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Wikipedia : In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational (or equivalently, integer) coefficients.

  • Question : Show that the Set of algebraic number in $\mathbb{C}$ is countable ?

  • Hint : if $\mathbb{F(z) = a_ 0 + a_1z + .... + a_nz^n}$ we denote the Height of a polynomial as : $\mathbb{h = |a_0| + |a_1| + ...... + |a_n| + n}$

PS : i have already solve it , i am looking for different way to solve it ;)

  • 2
    How big is the set of polynomials of a given degree with integer coefficients?2011-01-02
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    Enumerate the polynomials with integer coefficients.2011-01-02
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    You first need the fact that every algebraic number,$t$, corresponds to a "minimal polynomial". ie a monic polynomial with integer coeficients, having $t$ as a root, whose degree is as small as posible. Show that the set of such polynomials is countable.2011-01-02
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    Is this a homework assignment? I mean, I just wrote this exact question in a problem set yesterday.2011-01-02
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    @Joe: Why do you need that?2011-01-02
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    @Tony K. You right. We don't really need minimal polynomials The fact that $Z[x]$ is countable is good enough. I just find it easier to consider a specific polynomial for each algebraic number.2011-01-02
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    I deleted user5315's anser/hint, since he already duplicated that content in the question statement above.2011-01-03

1 Answers 1