## 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 $\mathbb{Q}$.