(Solved): Exercise 3. (26 points) For each part of this exercise, let \( p(x)-x^{4}-2 \) and let. \( \mathbb ...

Exercise 3. (26 points) For each part of this exercise, let \( p(x)-x^{4}-2 \) and let. \( \mathbb{K}-\mathbb{Q}(\sqrt[4]{2}, i) \), where \( \sqrt[4]{2} \) denotes the real root of \( p(x) \) (so \( \sqrt[4]{2} \in \mathbb{R} \) ). You may assume (without proof) that \( p(x) \) is irreducible over \( \mathbb{Q} \). (a) (10 points) Prove that \( [\mathbb{K}: \mathbb{Q}]-8 \). Note (you do not need to prove this) that the roots of \( p(x) \) are \[ \sqrt[4]{2}, \quad-\sqrt[4]{2}, \quad i \sqrt[4]{2}, \quad-i \sqrt[4]{2} . \] (b) (2 points) Prove that all of the roots of \( p(x) \) are in \( \mathbb{K} \). You must cite the field axioms and the definition of \( \mathbb{Q}(\sqrt[4]{2}, i) \).