Compute the degree of QQ(i + √2)

Compute the degree of K = \mathbb{Q}(i + \sqrt{2}) over \mathbb{Q}. Exhibit three distinct subfields of K. Deduce that the polynomial found in this previous exercise is irreducible over \mathbb{Q}.

Since we know \theta = i + \sqrt{2} is a root of p(x) = x^4 - 2x^2 + 9, the degree of \theta over \mathbb{Q} is at most 4. In this previous exercise, we showed that K contains both i and \sqrt{2}, so that \mathbb{Q}(i), \mathbb{Q}(\sqrt{2}) \subseteq K. Now i\sqrt{2} \in K, so we also have \mathbb{Q}(i\sqrt{2}) \subseteq K. Note that the minimal polynomial of i over \mathbb{Q} is a(x) = x^2 + 1 (irreducible because a(x+1) is Eisenstein at 2), the minimal polynomial of \sqrt{2} is b(x) = x^2 - 2 (Eisenstein at 2) and the minimal polynomial of i\sqrt{2} is c(x) = x^2 + 2 (Eisenstein at 2). So these subfields all have degree 2 over \mathbb{Q}.

Suppose \alpha,\beta \in \mathbb{Q} and (\alpha + i\beta)^2 = 2. Comparing coefficients, we see that \alpha^2 - \beta^2 = 2 and \alpha\beta = 0. If \alpha = 0, then \beta^2 = -2, a contradiction since \beta^2 \geq 0. If \beta = 0, then \alpha^2 = 2, a contradiction since \sqrt{2} is not rational. In particular, \sqrt{2} \notin \mathbb{Q}(i), so that \mathbb{Q}(i) and \mathbb{Q}(\sqrt{2}) are distinct.

Similarly, if (\alpha + \beta i\sqrt{2})^2 = 2, then either \alpha^2 = 2 or \beta^2 = -1. So \mathbb{Q}(\sqrt{2}) and \mathbb{Q}(i\sqrt{2}) are distinct.

Finally, if (\alpha + i\beta\sqrt{2})^2 = -1, then either \alpha^2 = -1 or \beta^2 = 1/2, again a contradiction. So \mathbb{Q}(i) and \mathbb{Q}(i\sqrt{2}) are distinct.

Now we claim that the degree of \sqrt{2} over \mathbb{Q}(i) is 2. To see this, note that (since \mathbb{Q}(i) and \mathbb{Q}(\sqrt{2}) are distinct) x^2-2 is irreducible over \mathbb{Q}(i), and so is the minimal polynomial of \sqrt{2} over \mathbb{Q}(i). So \mathbb{Q}(i,\sqrt{2}) = \mathbb{Q}(i+\sqrt{2}) has degree 2 over \mathbb{Q}(i); hence K has degree 4 over \mathbb{Q}.

We can summarize this information using the following labeled lattice diagram.

A lattice of fields.

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