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Math Help - need help with a proof if f(b) = 0 and b is not an integer, then b is irrational

  1. #1
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    need help with a proof if f(b) = 0 and b is not an integer, then b is irrational

    I do not even know where to start with this. this proof is for an abstract algebra course.

    Suppose f(x) = xn + a1xn-1 + . . . + a1x + a0 where the coefficients a0,a1, . . . ,an-1 are integers.
    prove: if f(b) = 0 and b is not an integer, then b is irrational.
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  2. #2
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    Re: need help with a proof if f(b) = 0 and b is not an integer, then b is irrational

    Thanks from HallsofIvy
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  3. #3
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    Re: need help with a proof if f(b) = 0 and b is not an integer, then b is irrational

    Thank you for the starting point!!!
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  4. #4
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    Re: need help with a proof if f(b) = 0 and b is not an integer, then b is irrational

    suppose

    f\left(\frac{p}{q}\right) = 0 with p,q integers with gcd(p,q) = 1.

    then

    p^n = q(-a_{n-1}p^{n-1}-\dots-q^{n-2}a_1p-q^{n-1}a_0).

    then q divides pn, but gcd(p,q) = 1, so gcd(pn,q) = 1.

    these two facts together imply q = 1, so p/q is an integer.
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