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Algebraic closure

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Show that the algebraic closure of Q (rational numbers) in C (complex numbers) (and hence any algebraic closure of Q, once we have the uniqueness statement) is countable.

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Solution Summary

This provides an example of proving that an algebraic closure is countable.

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It's enough to show that the set A of all algebraic numbers is countable (if one allows the coefficients of a polynomial
to be rational, we can always clear denominators to obtain one with integer coefficients, by multiplying by
the common denominator).

Let P be the collection of all polynomials with integer coefficients;
and for each natural n, let P_n ...

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