Question

Find quotient ring Q[x]/(x^4-10x*x+1), and find number field which it is isomorphic.

Answer 1

$$x ^ {4} - 10 x ^ {2} + 1 = 0$$ $$D = 100 - 4 = 96$$ $$x ^ {2} = \frac {10 \pm 4 \sqrt {6}}{2} = 5 \pm 2 \sqrt {6} = \left(\sqrt {3} \pm \sqrt {2}\right) ^ {2}$$ $$x = \sqrt {3} \pm \sqrt {2} \Rightarrow \mathbb {Q} [ x ] / \left(x ^ {4} - 10 x ^ {2} + 1\right) \cong \mathbb {Q} \left(\sqrt {2} + \sqrt {3}\right)$$