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Math Help - Finding minimal polynomial

  1. #1
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    Finding minimal polynomial

    Here's an easy question: What is the minimal polynomial of (1/3)+(1/6)(1+sqrt(-3))((1/2)(25-3sqrt(69)))^(1/3) + (1-sqrt(-3))/(3*2^(2/3)(25-3sqrt(69))^(1/3)

    (insert this into Mathematica or your favorite program to make sense of it. If there's something missing, it's a root of x^5+x+1)

    The minimal polynomial is, according to Mathematica, x^3-x^2+1, question is how to come to that conclusion? I've only seen this done with simple examples.
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  2. #2
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    Re: Finding minimal polynomial

    x^5+x+1 factors into irreducible polynomials (x^2+x+1)(x^3-x^2+1) over \mathbb{Q}. Your expression is not a root in the first one, so...
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    Re: Finding minimal polynomial

    Double checked the expression and it's correct except for missing the very last parenthesis, so, yes, it's a root of x^5+x+1.
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    Re: Finding minimal polynomial

    Quote Originally Posted by spudwish View Post
    Double checked the expression and it's correct except for missing the very last parenthesis, so, yes, it's a root of x^5+x+1.
    Sorry, what I meant was that your expression is not a root of the first factor x^2+x+1.
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  5. #5
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    Re: Finding minimal polynomial

    True, it's a root of the second factor, but I don't see how either of that is relevant to the question. (here's where I'm not sure how to relay that I'm really asking...)

    Perhaps I phrased my question poorly: Given the root above, how do I manually find its minimal polynomial? What I mean is a sequence of steps like this one:

    Minimal polynomial for a:=sqrt(2)+sqrt(3):

    x=sqrt(2)+sqrt(3)
    (x-sqrt(2))^2=sqrt(3)^2
    x^2-2xsqrt(2)+2=3
    x^2-1=2x sqrt(2)
    (x^2-1)^2 = (2x sqrt(2))^2
    x^4-2x+1=8x^2
    So minpoly(a) = x^4-10x+1
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  6. #6
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    Re: Finding minimal polynomial

    I'll work on that and update this post later. May I ask how you came by such an equation? The method I showed you was a good reason why Galois theory is invented...

    EDIT: Sorry, but I have no wish to work this out by hand. If you're looking for a purely mechanical way to do this with no intuition whatsoever, the computer is the way to go. However if you know the degree (or approximate range) of the minimal polynomial f(x), say degree n, you can calculate the explicit powers \alpha^2,\alpha^3,...\alpha^n. Then write f(\alpha)=\displaystyle{\sum_{i=0}^n c_i\alpha_i}=0 and solve for the coefficients c_i using a matrix and an appropriate basis depending on what \alpha is.
    Last edited by Gusbob; March 24th 2013 at 06:59 PM.
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  7. #7
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    Re: Finding minimal polynomial

    Alright thanks. I've been reading up on GT for a way overdue bachelor thesis, hence been doodling with polynomials quite a bit, I guess the question just popped up in my head how one would find the minimal polynomial for a more difficult radical expression than the ones featured in standard textbooks
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  8. #8
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    Re: Finding minimal polynomial

    I made some "conceptual" progress on my own: take the easier example of finding the minimal polynomial of a:=2^(1/3)+2^(2/3). To see if we're heading in the right direction, we use Mathematica to see that its minimal polynomial is x^3-6x-6. What next? We can subtract one or both terms from RHS, square or cube, or leave as is and square or cube. We'll cube as is. Put:

    x = 2^(1/3)+2^(2/3) (=a)
    x^3 = 6+6a = 6+6x.

    Hence the minimal polynomial of a is x^3-6x-6, like we wanted. I was a bit puzzled by this result, didn't it just produce a polynomial not in Q[x], I thought. But, the ansatz was explicitly that x=a, or basically to produce a polynomial "in a" over Q, of smallest degree >1, which we did. This particular example isn't very enlightening really, but the reasoning is the same for, say, b=2^(1/3)+3^(1/3); we'll start raising powers, and not until we hit degree 9 will be able to factorize the resultant polynomials so that it's a polynomial "in b".

    edit: Didn't see your edit there above. I'll check out your approach, thanks!
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