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I'm looking for references, if there is any, for this problem:

Characterize all elements $a \in M_n(\mathbb{Z})$ for which we have $\mathbb{Q}[a] \cap M_n(\mathbb{Z})=\mathbb{Z}[a].$

Here, by $C[a]$ I mean the ring of polynomials in $a$ with coefficients in a ring $C.$

Thank you

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    One direction is easy, that $\mathbb{Q}[a] \cap M_n(\mathbb{Z})$ contains $\mathbb{Z}[a]$ for any $a \in M_n(\mathbb{Z})$. So the question amounts to for which $a$ the left-hand expression doesn't yield any extra integer matrices. A necessary condition is that $a$ has entries with greatest common divisor $1$.2012-10-19
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    Let $a = \begin{pmatrix} 1&2 \\ 4&3 \end{pmatrix}$. Then $\mathbb{Q}[a]$ contains $\begin{pmatrix} 1&1 \\ 2&2 \end{pmatrix}$ but this is not in $\mathbb{Z}[a]$. Thus it is not sufficient that $a$ has entries with GCD $1$, although this is a necessary condition for $\mathbb{Q}[a] \cap M_n(\mathbb{Z})=\mathbb{Z}[a]$.2012-10-21
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    I don't know why you think that the gcd of entries of those $a$ that satisfy the property in the problem must be one? For example, if $I$ is the identity matrix and $a=mI, \ m \in \mathbb{Z},$ then $a$ satsifies the property in the problem, doesn't it?2012-10-22
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    Yes, that's true. Let me restate it as if $a \notin \mathbb{Z}[I]$, so that we have a simple proper extension, then the gcd of $a$'s entries must be 1 in order for the rational polynomials in $a$ not to produce an additional integer matrix.2012-10-22

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