Integral basis of a two-prime biquadratic field ($\mathbb{Q}(\sqrt{2},\sqrt{p}),p\equiv 3(\mathrm{mod}4)$)

For an odd prime $p\equiv 3(\mathrm{mod}4)$, traces to the three quadratic subfields show that an integer has coefficients $(a+b\sqrt{p}+c\sqrt{2}+d\sqrt{2p})/2$ with rational integers $a,b,c,d$. Its norm to $\mathbb{Q}(\sqrt{p})$ forces $a,b$ even and $c\equiv d(\mathrm{mod}2)$. Hence an integral basis is $1,\sqrt{2},\sqrt{p},(1+\sqrt{p})\sqrt{2}/2$. The last generator satisfies the monic polynomial $X^4-(p+1)X^2+(p-1{)}^2/4$, proving integrality as well as spanning. The subfield used for the norm matters: the norm to $\mathbb{Q}(\sqrt{2})$ has different coefficients.