Compute the conjugates of a given element over a field extension

Let \theta = \sqrt[4]{5}, \beta = \sqrt{5}, and K = \mathbb{Q}(\theta).

  1. Compute the conjugates of \theta.
  2. Compute the conjugates of \beta over K.
  3. Find the field polynomial for \beta over K, and express it as a power of the minimal polynomial of \beta over \mathbb{Q}.
  4. Find the conjugates of \gamma = 3 - 2\sqrt[4]{5} over K and compute the degree of \mathbb{Q}(\gamma) over \mathbb{Q}.

Note that \theta is a root of p(x) = x^4 - 5, which is irreducible by Eisenstein’s criterion. Evidently the roots of p(x) (i.e. the conjugates of \theta) are \pm \theta and \pm i \theta.

Certainly \beta = q(\theta), where q(x) = x^2. Now q(\theta) = q(-\theta) = \sqrt{5} and q(i\theta) = q(-i\theta) = -\sqrt{5} are the conjugates of \beta over K.

The field polynomial of \beta over K is then (x - \sqrt{5})^2(x + \sqrt{5})^2 = (x^2 - 5)^2; note that x^2 - 5 is the minimal polynomial of \beta over \mathbb{Q} (it is irreducible by Eisenstein.)

Note that \gamma = c(\theta), where c(x) = 3 - 2x. Evidently then the conjugates of \gamma over K are 3 \pm 2\theta and 3 \pm 2i\theta. Since \mathbb{Q}(\theta) has degree 4 over \mathbb{Q} and these conjugates are all distinct, then by Theorem 5.10 in TAN, \mathbb{Q}(\gamma) = \mathbb{Q}(\theta) has degree 4 over \mathbb{Q}.

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