= Laurent normalization by a monomial change of variables
{c}
{title2=$X_j=Y_jY_r^{N^j}\ (j<r)$}
A nonzero quotient of an $r$-variable <Laurent polynomial ring> is finite over an embedded <Laurent polynomial ring> in $m\leq r$ variables. Given a nonzero Laurent relation, choose $N$ so that weights $(N,N^2,\ldots,N^{r-1},1)$ separate its finite exponent support. The displayed invertible <monomial> substitution makes its extreme $Y_r$ coefficients units in the remaining Laurent variables. Normalize the relation to a <monic polynomial> with unit constant term; both $Y_r$ and $Y_r^{-1}$ are integral over the image of the $(r-1)$-variable subring. Induction and transitivity of finite extensions prove the assertion, over finite <fields> as well as infinite <fields>. An infinite-dimensional quotient has $m\geq1$.
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