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I’ve seen usage of the notation $\mathbb Z[x]$ before, where $x$ is some non-integer. For example in the Wikipedia article on the Gaussian integers it says that the set of Gaussian integers can be written as $\mathbb Z[i]$. But what does it actually mean?

I suspect that $\mathbb Z[x]$ refers to the set of numbers of the form $a+bx$, where $a,b\in\mathbb Z$, but I’m not sure.

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$\mathbb Z[x]$ is the ring of all polynomials over the variable $x$ with integer coefficients. So this includes $a+bx$ (for all $a,b \in \mathbb Z$), but also things like $x^2$, and the other higher order polynomials.

Whereas $\mathbb Z[i]$ is what we get when we evaluate the polynomials at $i$. That is, we take an integer polynomial $f(x)$ and plug in $i$ for $x$ to get $f(i)$, a complex number which we can always write in the form $a +bi$ for some integers $a,b$.

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