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answered divisible by all standard prime numbers
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comment Algorithm for determining when polynomial iteration is bounded?
@Per: Thank you for the reference to the result on IFS undecidability. Alas, I don't see how to the IFS result here. However the second point, about the map $x\to x-1/x$ seems really interesting. Do you have a reference?
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asked Algorithm for determining when polynomial iteration is bounded?
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comment The existence of non-trivial homomorphisms $\prod_{n=1}^{\infty}\mathbb{Z}/\bigoplus_{n=1}^{\infty}\mathbb{Z}\to\mathbb{Z}/p\mathbb{Z}$
Nice. Alternatively, we can think of the last displayed item as a ring $R$, every element of which satisfies the equation $x^p=x$. So if $I$ is any maximal ideal of $R$, then the elements of the field $R/I$ all satisfy the same equation, whence $R/I$ is the $p$-element field.
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answered The existence of non-trivial homomorphisms $\prod_{n=1}^{\infty}\mathbb{Z}/\bigoplus_{n=1}^{\infty}\mathbb{Z}\to\mathbb{Z}/p\mathbb{Z}$
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comment A question about how polynomials simplify under substitution
The sequence of revisions in this post is confusing, and may have misled you, for which I am sorry. Note that in the example you give, the group $M_0$ contains the nonconstant polynomial $x_1-x_0$.