Compute the splitting field of x⁴-2 over QQ

Compute the splitting field of p(x) = x^4-2 over \mathbb{Q}, as well as its degree.


Let \sqrt[4]{2} denote the positive real fourth root of 2. Evidently, p(x) = (x-\sqrt[4]{2})(x+\sqrt[4]{2})(x-i\sqrt[4]{2})(x+i\sqrt[4]{2}). So the splitting field of p(x) over \mathbb{Q} is \mathbb{Q}(\sqrt[4]{2}, i\sqrt[4]{2}) = \mathbb{Q}(i,\sqrt[4]{2}).

Note that \mathbb{Q}(i) has degree 2 over \mathbb{Q}. Over the UFD \mathbb{Z}[i], p(x) is Eisenstein at 1+i, hence irreducible, and so by Gauss’ Lemma, is irreducible over \mathbb{Q}(i) (the field of fractions of \mathbb{Z}[i]). So \mathbb{Q}(i,\sqrt[4]{2}) has degree 4 over \mathbb{Q}(i). We can visualize this scenario in the following diagram.

A field diagram

So \mathbb{Q}(i,\sqrt[4]{2}) has degree 8 over \mathbb{Q}.

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