Algebraic numbers proof

Any hints on how I can start this?? I know that algebraic numbers are those which satisfy a polynomial equaling 0 but I'm not sure how to use it here at the moment.

Start with the definition of an algebraic number.

So, x is a root of some polynomial p(x)

How might you tweek that polynomial so that root(x) is a root?

So $a_1x^n+a_2x^{n-1}+a_3x^{n-2}+...+a_{n-1}x+a_n=0$ is true.

Would I do the mapping $x \mapsto \sqrt{x}$? It's what I had initially but it doesn't seem rigorous enough so I dropped it, can't see anything else other than that.

I suspect you're on the right lines, but I don't know what you mean by do the mapping here.

Perhaps I worded it wrong, my mind is going blank today all over

I meant to say that I can map all $x>0$ solutions onto their roots, so all x values transfer over to some function of itself, in this case the square root. This gives $a_1(\sqrt{x})^n+a_{2}(\sqrt{x})^{n-1}+...+a_{n-1}(\sqrt{x})+a_n=0$ which would show that $\sqrt{x}$ is algebraic since it satisfies the polynomial.

Any hints on how I can start this?? I know that algebraic numbers are those which satisfy a polynomial equaling 0 but I'm not sure how to use it here at the moment.

OK, I follow you, but you've gone the wrong way - you don't know root(x) is a root of that polynomial.

Here's an example:


The simpliest polynomial with 2 as a root is:
$x-2=0$

The simpliest poly. with root(2) as a root is:

$x^2-2=0$

How do you get the latter from the former.

Got to go out now.

Unless I'm missing something (possible, I'm a bit thick tbh), this seems fairly straightforward.
x^n = (x^(1/2))^(2n).

I think that's exactly it since from that procedure you get another polynomial with root x as the solution. Can't believe something so simple evaded me so longer than it should've

As I understand it going the other way constructively (finding the actual poly) is quite hard.

But if x is algebraic then [Q(x):Q] is finite, therefore so is [Q(x^2):Q] so you know a poly exists without haviNg to find it.

Does this approach fully prove it?? It is not finding any particular polynomials but it turned out more long-winded than I thought it would be. Thanks!

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