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Formal Derivative and Multiple Roots

  1. Feb 10, 2013 #1
    Primitive roots of 1 over a finite field

    1. The problem statement, all variables and given/known data

    The polynomial x3 − 2 has no roots in F7 and is therefore irreducible in F7[x]. Adjoin a root β to make the field F := F7(β), which will be of degree 3 over F7 and therefore of size 343. The
    multiplicative group F× is of order 2 × 9 × 19. Show:
    (1) that −1 is a primitive square root of 1 in F;
    (2) that β is a primitive 9th root of 1 in F;
    (3) that −1 + β is a primitive 19th root of 1 in F;
    (4) and then that β − β2 is a primitive root in F, that is, a generator of the multiplicative group F×.

    2. Relevant equations



    3. The attempt at a solution

    I am totally confused about how to even start this proof. Any help would be greatly appreciated please!
     
    Last edited: Feb 10, 2013
  2. jcsd
  3. Feb 11, 2013 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    β3-2=0
    or, equivalent, β3+5=0
    You can add and multiply polynomials with β as usual, just consider everything mod 7, and use that equation to keep the degree of your polynomials smaller than 3.

    Therefore, -1 = 6 (as 6+1=0), and 6*6=...
    You can calculate β, β^2 (well, nothing to do for those), β^3, β^4, ... and show that it is a primitive 9th root in that way.
    (3) would be possible like that, too, but there might be a more elegant way.
     
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