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Extending a field by a 16th primitive root of unity

  1. Nov 23, 2016 #1
    1. The problem statement, all variables and given/known data
    let c be a primitive 16th root of unity. How many subfields M<Q(c) are there such that [M:Q] = 2

    2. Relevant equations


    3. The attempt at a solution
    I think the only subfield M of Q(c) such that [M:Q] = 2 is Q(c^8). Then M = {a+b(c^8) such that a,b are elements of Q}. I'm thinking about the other powers of c and trying to think if any other would generate an extension field over Q with a degree of 2. Any number that's relatively prime to 16 would be another primitive 16th root of unity, so we can throw out all odd numbers. Q(c^2) and Q(c^14) would both be degree 8, Q(c^4) and Q(c^12) would both be degree four, Q(c^6) would be of degree 8 and Q(c^10) would also be of degree 8. So it's only [Q(c^8):Q]= 2 correct?
     
  2. jcsd
  3. Nov 23, 2016 #2

    andrewkirk

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    Think again about that '8' exponent. Note that ##c^8=-1##, which is in ##\mathbb Q##, so ##\mathbb Q(c^8)=\mathbb Q## (ie the extension is trivial), whence ##[\mathbb Q(c^8):\mathbb Q]=1##, not 2.
    Why? I think you are confusing the degree of the extension ##[\mathbb Q(c^k):\mathbb Q]##, which is the dimension of the vector space, with the order of the element ##c^k## in the multiplicative group ##\{c^j\ :\ j\in\{0,1,...,15\}\}##.
     
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