Undergrad Understanding Algebraic Numbers & Proving Them

[swampwiz]

I'm trying to grok what an algebraic number could look like. Yes, I understand that an algebraic number is any number that could be a solution (root) to a polynomial having integer coefficients (or rational coefficients, since any set of rational coefficients can be made into integers by scaling the entire polynomial equation).

I can't prove it, but it seems that an algebraic number follows a few rules:

- any rational number is an algebraic number

- any deMoivre root of an algebraic number is an algebraic number

- the sum or product of any pair of algebraic numbers is an algebraic number

Is my understanding accurate here? If so, is there any way to prove this?

 

[Infrared]

Yes, these are all true.

- The polynomial $x-(a/b)$ has $a/b$ as a root.

- If $\alpha$ is a root of $p(x)$ and if $w^n=\alpha$, then $w$ is a root of $p(x^n).$

- If $\alpha$ and $\beta$ are algebraic, then $[\mathbb{Q}(\alpha,\beta):\mathbb{Q}]=[\mathbb{Q}(\alpha,\beta):\mathbb{Q}(\alpha)]\cdot [\mathbb{Q}(\alpha):\mathbb{Q}]$ is finite, so $\alpha\beta,\alpha+\beta\in\mathbb{Q}(\alpha,\beta)$ are algebraic.

The point in the last argument is that $[\mathbb{Q}(\gamma):\mathbb{Q}]$ is finite if and only if $\gamma$ is algebraic. It's not easy to explicitly describe the minimal polynomial of $\alpha+\beta$ in terms of the minimal polynomials of $\alpha$ and $\beta.$