Geodesic
Representation of the direct sum of two quadratic fields by rational symmetric matrices
Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 4, Tome 67 (1977), pp. 195-200

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Let $f$ be a fourth-degree polynomial over the field of rational numbers $\mathbf Q$ with leading coefficient $I$ which decomposes over $\mathbf Q$ into the product of two irreducible second-degree polynomials. It is proved that in order that $f$ be the characteristic polynomial of a symmetric matrix with elements in $\mathbf Q$, it is necessary and sufficient that all the roots of $f$ be real. 
@article{ZNSL_1977_67_a10,
     author = {L. I. Roginskii},
     title = {Representation of the direct sum of two quadratic fields by rational symmetric matrices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {195--200},
     publisher = {mathdoc},
     volume = {67},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1977_67_a10/}
}
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L. I. Roginskii. Representation of the direct sum of two quadratic fields by rational symmetric matrices. Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 4, Tome 67 (1977), pp. 195-200. http://geodesic.mathdoc.fr/item/ZNSL_1977_67_a10/