Compute powers in a field extension

Show that p(x) = x^3+x+1 is irreducible over \mathbb{F}_2, and let \theta be a root. Compute the powers of \theta in \mathbb{F}_2(\theta).


Note that p(x) has no roots in \mathbb{F}_2, and since p(x) has degree 3, it is thus irreducible.

In \mathbb{F}_2(\theta), we have \theta^3 = \theta+1. Thus the powers of \theta are as follows.

  1. \theta^1 = \theta
  2. \theta^2 = \theta^2
  3. \theta^3 = \theta+1
  4. \theta^4 = \theta^2+\theta
  5. \theta^5 = \theta^2 + \theta + 1
  6. \theta^6 = \theta^2 + 1
  7. \theta^7 = 1
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