Finding an irreducible polynomial over the rationals

Question

I'm very confused by a homework question. "Find the irreducible polynomial for $ \sin{2\pi/5}$ over Q.
I found that $16t^{4}-20t^{2}+5=0$ but this is not monic? This is also irreducible by Eisenstein, but minimal polynomials are always monic?

Answer 1

If you want a monic polynomial, you can divide by $16$. Of course it won't have integer coefficients any more. A number that is a root of a monic polynomial with integer coefficients is an algebraic integer. $\sin(2\pi/5)$ is not an algebraic integer.

Answer 2

$p(t)=t^{4}-5/4t^{2}+5/16$ is the minimal polynomial of $\sin 2\pi/5$ over $\mathbb Q$. It is irreducible in $\mathbb Q[x]$ according to Gauss's lemma as $16t^{4}-20t^{2}+5$ is irreducible in $\mathbb Z[x]$.