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An extended comment on Torsten's reasonable answer: It might be interesting to know why these special Jordan blocks are singled out at first. I guess these are (in old-fashioned language) Jordan forms of "nonderogatory" nilpotent matrices as treated in matrix algebra books (characteristic polynomial = minimal polynomial)? Anyway, there is quite a bit of older literature in this direction including a 1972 Academic Press book Integral Matrices by Morris Newman which integrates some group theory with linear algebra in the spirit of Fricke-Klein.

I'd approach the basic question here concretely by observing first that the matrices with rational entries similar to the given Jordan block are precisely the matrices over $\mathbb{Q}$ having this Jordan form. For this the point is to compare the rational canonical form over $\mathbb{Q}$ with the Jordan form over $\mathbb{C}$: here there is no problem with the eigenvalues of a given nilpotent matrix being in the smaller field. In turn, the similarity of an integral matrix with its Jordan form over $\mathbb{Q}$ can just as easily be carried out using an invertible integral matrix (after factoring out a common denominator for fractions involved in a given similarity matrix)runs into further problems of an arithmetic nature which I don't know how to analyze so concretely.

In the broader Lie-theoretic setting, you have the orbit of a regular nilpotent element in the full matrix algebra under the adjoint group associated to the general linear group. Here there are interesting arithmetic questions, when you start with an orbit over (say) $\mathbb{Q}$ and try to relate it to orbits under restriction to a natural arithmetic subgroup (here the group over $\mathbb{Z}$). There is substantial literature on all of this, related to features of a number field and its ring of integers, which I don't have at my fingertips. But that's probably the most natural setting for the question asked. The modern literature on arithmetic subgroups of reductive groups is extensive and beautiful.

2 added 710 characters in body

An extended comment on Torsten's reasonable answer: It might be interesting to know why these special Jordan blocks are singled out at first. I guess these are (in old-fashioned language) Jordan forms of "nonderogatory" nilpotent matrices as treated in matrix algebra books (characteristic polynomial = minimal polynomial)? Anyway, there is quite a bit of older literature in this direction including a 1972 Academic Press book Integral Matrices by Morris Newman which integrates some group theory with linear algebra in the spirit of Fricke-Klein.

I'd approach the basic question here concretely by observing first that the matrices with rational entries similar to the given Jordan block are precisely the matrices over $\mathbb{Q}$ having this Jordan form. For this the point is to compare the rational canonical form over $\mathbb{Q}$ with the Jordan form over $\mathbb{C}$: here there is no problem with the eigenvalues of a given nilpotent matrix being in the smaller field. In turn, the similarity of an integral matrix with its Jordan form over $\mathbb{Q}$ can just as easily be carried out using an invertible integral matrix (after factoring out a common denominator for fractions involved in a given similarity matrix).

In the broader Lie-theoretic setting, you have the orbit of a regular nilpotent matrix element in the full matrix algebra under the adjoint group associated to the general linear group. Here there are interesting arithmetic questions, when you start with an orbit under over (say) $\mathbb{Q}$ and try to relate it to orbits under restriction to a natural arithmetic subgroup (here the group over $\mathbb{Z}$). There is substantial literature on all of this, related to features of a number field and its ring of integers, which I don't have at my fingertips. But that's probably the most natural setting for the question asked. The modern literature on arithmetic subgroups of reductive groups is extensive and beautiful.

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An extended comment on Torsten's reasonable answer: It might be interesting to know why these special Jordan blocks are singled out at first. I guess these are (in old-fashioned language) Jordan forms of "nonderogatory" nilpotent matrices as treated in matrix algebra books (characteristic polynomial = minimal polynomial)? Anyway, there is quite a bit of older literature in this direction including a 1972 Academic Press book Integral Matrices by Morris Newman which integrates some group theory with linear algebra in the spirit of Fricke-Klein.

In the broader Lie-theoretic setting, you have the orbit of a regular nilpotent matrix in the full matrix algebra under the adjoint group associated to the general linear group. Here there are interesting arithmetic questions, when you start with an orbit under (say) $\mathbb{Q}$ and try to relate it to orbits under restriction to a natural arithmetic subgroup (here the group over $\mathbb{Z}$). There is substantial literature on all of this, related to features of a number field and its ring of integers, which I don't have at my fingertips. But that's probably the most natural setting for the question asked. The modern literature on arithmetic subgroups of reductive groups is extensive and beautiful.