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Finding the minimal polynomial of an irrational over Q

  1. Oct 28, 2016 #1
    1. The problem statement, all variables and given/known data
    Let a = (1+(3)^1/2)^1/2. Find the minimal polynomial of a over Q.

    2. Relevant equations


    3. The attempt at a solution
    Maybe the first thing to realize is that Q(a):Q is probably going to be 4, in order to get rid of both of the square roots in the expression. I also suspect that +/-(3)^1/2 will be roots of the minimal polynomial, as Q(a):Q = [Q(a):Q((3)^1/2)]*[Q(3^1/2):Q]. I do not know where to go from here, any advice PF?
     
  2. jcsd
  3. Oct 28, 2016 #2

    lurflurf

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    Homework Helper

    $$a=\sqrt{1+\sqrt{3}}$$
    so your on the right track looking for 4 conjugates
    consider all sign variations of square roots
    $$\pm\sqrt{1\pm\sqrt{3}}$$
    the minimum polynomial will be
    (x-a)(x-b)(x-c)(x-d)
    where a,b,c,d are the four conjugates
    $$a=\sqrt{1+\sqrt{3}}\\
    b=\sqrt{1-\sqrt{3}}\\
    c=-\sqrt{1-\sqrt{3}}\\
    d=-\sqrt{1+\sqrt{3}}\\$$
    another possibly easier approach is to isolate 3 in your equation for a
    the minimal polynomial of 3 is
    x-3
    so
    f(a)-3
    is the minimal polynomial of a if f(a) is 3 in terms of a
     
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