Question: A number is said to be algebraic if it is the root (zero) of a polynomial with integer coefficients. Show that the set of algebraic numbers is countable.

Question

usukidoll · Answer

By Lemma 7.3.13, the set $Z[x]$ of all polynomial functions with integer coefficients is countable. By Theorem 7.3.15, the set of algebraic numbers is countable. If $p$ is a polynomial function, then a number $z$ is a root of $p$ if $p(x)=0$. For example, the roots of the function $x^3-3x-2$ are $-1$ and $-2$. The number of roots of any polynomial function is finite. Since the set $Z$ is countable, the Cartesian product of finitely many copies of $Z$ is a countable set. It follows that each set $P_n$ is also a countable set.

usukidoll · Answer

sigh don't know what happens next..

usukidoll · Answer

I get stuck afterwards...

usukidoll · Answer

I'm not even sure if this is correct... x.x

anonymous · Answer

Okay, so here is my idea...

anonymous · Answer

If you can map to a vector with $n$ elements, then you can map to a degree $n$ polynomial.  Just let the vector components be the coefficients.

anonymous · Answer

Basically: $$
f(a_0,a_1,a_2,\ldots , a_n) = \sum_{k=0}^{n} a_kx^k
$$

usukidoll · Answer

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