Let $\displaystyle \color{red}a$ be an algebraic number.

Show that there is a single polynomial for $\displaystyle \color{red}a$ in $\displaystyle Q[x]$ of minimal order that has $\displaystyle \color{red}a$ as a root. This polynomial is called minimal polynomial of $\displaystyle \color{red}a$

Show that every polynomial that has $\displaystyle \color{red}a $ as a root, is divisible by the minimal polynomial of $\displaystyle \color{red}a$

Show that the minimal polynomial of $\displaystyle \color{red}a$ is irreducible in $\displaystyle Q[x]$

Thank you very much in advance