1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    2017 Award

    Staff: Mentor

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook