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Math Help - Find a monic polynomial, deduce that the polynomial is irreducible...

  1. #1
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    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.
    Last edited by TimsBobby2; November 25th 2012 at 09:37 AM.
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  2. #2
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    Re: Find a monic polynomial, deduce that the polynomial is irreducible...

    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)
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