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Math Help - Find the minimal polynomial for a = π^2 over Q(π^3)

  1. #1
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    Find the minimal polynomial for a = π^2 over Q(π^3)

    If it were in Q then the polynomial would be p(x) = x - π^2, but I don't know if being over Q(π^3) changes that. I think that when you compare the fields you get degree 1, but again, I'm not completely sure.
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  2. #2
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    Re: Find the minimal polynomial for a=π^2 over Q(π^3)

    If the extension is \mathbb Q(\pi)/\mathbb Q\left(\pi^3\right) the minimal polynomial is x^3-\pi^6.
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    Re: Find the minimal polynomial for a = π^2 over Q(π^3)

    Is this because the minimal polynomial needs to be degree three?
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    Member Sylvia104's Avatar
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    Re: Find the minimal polynomial for a=π^2 over Q(π^3)

    The minimal polynomial here is the monic polynomial f(x) of minimal degree with coefficients in \mathbb Q\left(\pi^3\right) such that f\left(\pi^2\right)=0. If f(x) has degree 1 or 2, f\left(\pi^2\right) is of the form a_0+\pi^2 or a_0+a_1\pi^2+\pi^4, where a_0,a_1\in\mathbb Q\left(\pi^3\right); clearly these cannot be 0.
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