How to show a number is irrational with this approach?

I read on MathOverflow the following:

"The number of proofs that we have of showing some numbers are irrational are very limited. We either show a number $α$ is irrational because it is algebraic of degree greater than one (by exhibiting an irreducible polynomial $f$ of degree greater than one $f(α)=0$)."

An algebraic number is a number that is a root for a polynomial with integers coefficients. What is an algebraic number of degree greater than one?

How do we show an "irreducible polynomial $f$ of degree greater than one $f(α)=0$" (a non-constant polynomial such that $\alpha$ is a root for it?)

Is there some literature that show that some number is irrational with this approach? Maybe a link on the internet or a book that show exactly this?