Thread: Find a monic polynomial, deduce that the polynomial is irreducible...

Set d = sqrt (3) + sqrt (2) in the real numbers R.

(a) Find a monic polynomial f(x) in Q[x] of degree 4 that has d as a root.
(b) If h, k, l, and m are rational numbers such that hd³+kd²+ld+m=0, substitute sqrt(3) + sqrt(2) for d, expand the result, and collect terms. Deduce that h,k,l, and m are all 0.
(c) Deduce from (b) that the polynomial f(x) in (a) is irreducible in Q[x].

Any help??

I was given this hint, HINT: Note that d - sqrt(3) = sqrt(2) so ( d - sqrt (3) )²​ = 2.

The four roots of the degree 4 poly are sqrt (3) + sqrt (2), sqrt (3) - sqrt (2), -sqrt (3) + sqrt (2), -sqrt (3) - sqrt (2). Write the equation with the roots as them. The square root terms with cancel out. This will answer (a)