= Integral basis of a two-prime biquadratic field
{title2=$\mathbb Q(\sqrt2,\sqrt p),\quad p\equiv3\pmod4$}
For an odd prime $p\equiv3\pmod4$, traces to the three quadratic subfields show that an integer has coefficients $(a+b\sqrt p+c\sqrt2+d\sqrt{2p})/2$ with rational integers $a,b,c,d$. Its norm to $\mathbb Q(\sqrt p)$ forces $a,b$ even and $c\equiv d\pmod2$. Hence an <integral basis> is $1,\sqrt2,\sqrt p,(1+\sqrt p)\sqrt2/2$. 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 $\mathbb Q(\sqrt2)$ has different coefficients.
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