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We say that a linear form $f=ax+by+cz$ of real coefficients is "irrational" if $a,b,c$ are linearly independent over the rationals. My question is: Are there 3 such "irrational" linear forms $f_1,f_2,f_3$ such that the product $f_1f_2f_3$ is of integral coefficients? Note that for the "2-dimensional" analogue, we have $(x+\sqrt{2}y)(x-\sqrt{2}y)=x^2-2y^2$. |
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This should be a comment to Robin's answer. Take any irreducible polynomial $f \in \mathbb{Q}[x]$ of degree 3 with real roots, say $\alpha, \beta, \gamma$. Set $f_1 = x + \alpha y + \alpha^2 z$, $f_2 = x + \beta y + \beta^2 z$, $f_3 = x + \gamma y + \gamma^2 z$. You can find plenty of polynomials here. |
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How about $x+2\cos(2\pi/7)y+4\cos^2(2\pi/7)z$, $x+2\cos(4\pi/7)y+4\cos^2(4\pi/7)z$, $x+2\cos(6\pi/7)y+4\cos^2(6\pi/7)z$ ? |
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