Unit of a Laurent polynomial ring ($(R[t,t^{-1}]{)}^{\times }=\{u{t}^k:u\in R^{\times },k\in \mathbb{Z}\}$)

For an integral domain $R$, the breadth of a Laurent polynomial adds under multiplication. If two Laurent polynomials multiply to one, both therefore have breadth zero: they are monomials, and their coefficients must be units of $R$. Conversely every displayed monomial is invertible. For $R=\mathbb{Z}$ the units are precisely $\pm t^k$, explaining the normalization ambiguity of an Alexander polynomial.