= Integral basis of the fourth-root-of-two field
{title2=$\mathcal O_{\mathbb Q(\sqrt[4]2)}=\mathbb Z[\sqrt[4]2]$}
Write $\alpha=\sqrt[4]2$ and an algebraic integer as $a+b\alpha+c\alpha^2+d\alpha^3$ with rational coefficients. Traces to $\mathbb Q(\sqrt2)$ show $2a,2b,2c,4d$ are integers. The norm has coefficients $a^2+2c^2-4bd$ and $2ac-b^2-2d^2$ in the integral basis $1,\sqrt2$. Set $A=2a$, $B=2b$, $C=2c$, $D=4d$. Integrality gives $A^2+2C^2-2BD\equiv0\pmod4$ and $4AC-2B^2-D^2\equiv0\pmod8$. The second forces $D$ even and then $B$ even; the first forces $A,C$ even; returning to the second forces $D$ divisible by four. Thus all original coefficients are integers, proving the integral basis $1,\alpha,\alpha^2,\alpha^3$.
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