# Polynomial of degree 4

1. Feb 11, 2007

### happyg1

1. The problem statement, all variables and given/known data

Let R be the field of real numbers and Q the field of rational numbers. In R,$$\sqrt 3$$ and $$\sqrt 2$$ are both algebriac. Exhibit a polynomial of degree 4 staisfied by$$\sqrt 2 + \sqrt 3$$.

2. Relevant equations

3. The attempt at a solution

I attempted to construct this by $$(\sqrt 2 + \sqrt 3)^4 = 20\sqrt 6 + 49$$
then my polynomial is $$x^4-(20\sqrt 6 + 49)=0$$
is this even close? ThenI have to show it's irreducible and I'm not quite sure where to start that.
My next problem is to prove that sin 1 is an algebriac number. I'm not sure how to construct that polynomial to prove it's irreducible, either.

Any hints or advise will be appreciated
Thanks
CC