I have been trying to see the properties of Minimal Polynomials. So from the examples I guessed the following properties but I am not sure whether they are true. The properties are:
1) If $\alpha$ is an algebraic number of degree $n >1$ and odd. The minimal polynomial has only one real root. 2) If $\alpha$ is an algebraic number of degree $n>1$ then all its roots are distinct.$\hspace{45mm}$ 3) If $\alpha$ is an algebraic number of degree $n>1$ then it does not have any rational root.
These are just some guesses. So it would be great,if anyone could point what are true and what are not and provide some hints. Thanks
I am completely new to all this I do not have any background in Galois theory. So it would be helpful if the answers use some elementary algebra.Also if some one can provide a reference it would be great.
Your second and third points are correct.
The second point. If $\mu\in\Bbb Q[X]$ is the minimal polynomial of $\alpha$ algebraic it has only simple roots since it must be irreducible. For if it did have a multiple root $\beta$, then $\mu$ and its derivative $\mu'$ would have a non trivial greatest common divisor in $\Bbb C[X]$ (at least $X-\beta$), say $$d=\text{gcd}_{\Bbb C[X]}(\mu,\mu')\in\Bbb C[X].$$ But both $\mu$ and $\mu'$ are rational polynomials, and the euclidean algorithm that calculates their greatest common divisor stays inside $\Bbb Q[X]$, so that $d$ is actually rational itself : $d\in\Bbb Q[X]$. But then $\mu$ has a non trivial factor $d\in\Bbb Q[X]$ (of degree $\leq \deg(\mu')<\deg(\mu)$) contradicting its irreducibility in $\Bbb Q[X]$.
The third point. Suppose the minimal polynomial $\mu$ of $\alpha$ algebraic (non rational) had a rational root $r\in\Bbb Q$. If $\mu=\nu (X_r)$ has $\nu\in\Bbb Q[X]$ and $\nu(\alpha)(\alpha-r)=0$, so $\nu(\alpha)=0$ (since by hypothesis $\alpha$ isn't rational, and so $\alpha\neq r$). This contradicts the minimality of $\mu$.