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Would the Smith Normal Form of the following matrix over $\mathbb Q[x]$

$$\begin{pmatrix}   (x+a)(x+b) & 0 & 0 &0 \\  0 & (x+c)(x+d) & 0 & 0 \\   0  &0 & x^3(x+a) & 0  \\   0 & 0 & 0& x^2(x+b)  \end{pmatrix}$$

 simply be

$$\begin{pmatrix}   f(x) & 0 & 0 &0 \\  0 & f(x) & 0 & 0 \\   0  &0 & f(x) & 0  \\   0 & 0 & 0& f(x)  \end{pmatrix}$$

where $f(x)= x^3(x+a)(x+b)(x+c)(x+d)$?

I am not sure because that would make the question quite trivial.

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    The Smith normal form for a matrix has to have the same determinant as the original matrix (up to multiplication by a unit), and your suggested form does not.2012-03-15
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    @GeoffRobinson: Right, thanks. Is there an effective way of finding the SNF?2012-03-15
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    Typically Gaussian elimination is used for finding SNF [Wikipedia page on SNF](http://en.wikipedia.org/wiki/Smith_normal_form) is a good pointer.2012-03-15

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