I'm trying to characterize the units of the polynomial ring $\mathbb{Z}[x, x^{-1}]$. I want to show that the units of this ring are exactly the elements $\pm x^n$, where $n$ is any integer. It is clear that all such elements are units of this ring, but I'm having trouble proving the converse (assuming that it's true).

It doesn't seem like an argument using degrees will work, because it's not even clear how to define the degree of a polynomial with integer exponents (e.g. what would be the degree of $x^5 + x^{-5}$?), and even if we could define a suitable notion of degree, there are units of $\mathbb{Z}[x, x^{-1}]$ of arbitrarily high 'degree' (so we can't rely on the fact that the degree of a unit is always $1$).

Can anyone give me some further hints or suggestions?

  • 2
    $\begingroup$ There are probably more elegant ways, but consider $p(x)q(x,x^{-1})=1$. By multiplying through by a suitable $x^N$, you get $p(x)\tilde q(x) = x^N$. Now use the fact that $\Bbb Z[x]$ is a UFD. $\endgroup$ – Ted Shifrin Jan 13 '17 at 1:37


Elements in this ring are Laurent's polynomials. Simply factor the lowest degree monomial (which is a unit): the quotient is an ordinary polynomial, which also has to be a unit. Which polynomials do you know to be units?


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