Compute the splitting field of x⁴+x²+1 over QQ

Compute the splitting field of p(x) = x^4 + x^2 + 1 over \mathbb{Q}, and its degree.


Note that p = g \circ h, where g(y) = y^2+y+1 and h(x) = x^2. Evidently, the roots of g(y) are \eta = \dfrac{-1 \pm i\sqrt{3}}{2}, and \zeta is a root of p if h(\zeta) = \eta. This yields the (distinct) roots \zeta = \pm \sqrt{\dfrac{-1 \pm i\sqrt{3}}{2}}. There are four of these, and so we have completely factored p.

The splitting field of p(x) over \mathbb{Q} is thus \mathbb{Q}(\pm \sqrt{\dfrac{-1 \pm i\sqrt{3}}{2}}) = \mathbb{Q}(\sqrt{-1/2 \pm \sqrt{-3/4}}).

Using this previous exercise, we have \sqrt{-1/2 + \sqrt{-3/4}} = 1/2 + i\sqrt{3}/2 and \sqrt{-1/2 - \sqrt{-3/4}} = 1/2 - i\sqrt{3}/2.

Evidently, then, the splitting field of p(x) is \mathbb{Q}(i\sqrt{3}). Since i\sqrt{3} is a root of the irreducible x^2+3 (Eisenstein), this extension of \mathbb{Q} has degree 2.

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