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Cyclotomic polynomials and primitive roots of unity

  1. Dec 4, 2006 #1
    w_{n} is primitive root of unity of order n, w_{m} is primitive root of unity of order m,
    all primitve roots of unity of order n are roots of Cyclotomic polynomials
    phi_{n}(x) which is a minimal polynomial of all primitive roots of unity of order n ,
    similarly, phi_{m}(y) is a minimal polynomial of all primitive roots of unity of order m ,
    then, what is the minimal polynomial of (W_{n},w_{m}), if exists or no????

    Thank you very much!! what book I can find some subject about primitve roots of unity.:wink:
     
  2. jcsd
  3. Dec 4, 2006 #2

    mathwonk

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    are you asking for the minimal polynomial of a product of two roots of 1?

    or of the field they generate together?

    and I presume you are working over the rationals Q, since yiou assume the cyclotomic polynomials are irreducible.


    i like van der waerden's old modern algebra, for a basic introduction. probably Gauss's disquisitiones is one of the best sources, but most number theoiry and abstract algebra books will say something, like hungerford, dummit - foote, michael artin, jacobson.
     
  4. Dec 5, 2006 #3
    I want to know the minimal polynomial of the field they generate together?

    Thank you !!!:wink:
     
  5. Dec 10, 2006 #4

    mathwonk

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    have you tried an example? for instance suppose z is a primitive complex 6th root of unity and w is a primitive complex 15th root of unity. then together they belong to the group of lets see 30th roots of unity. If you take say z^3, you get hmmmm -1 i suppose. anyway, it looks as if -w is a primitive 30th root of unity. is that right? then -w^5 sems like a primitive 6 th root of unity, and w a primitive 15th root.

    so it seems that together they generate the same field as w does alone.


    try another one, like 18th root and 12th root. what happens?
     
  6. Dec 10, 2006 #5

    mathwonk

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    by the way there is no such thing as"the minimal polynomial of a field", you need to choose a generating element first, but here you can always choose it to be another primitive root of unity, perhaps of roder the lcm of the two orders you start with?
     
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