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6 2. Non-singularity and Resolution of Singularities as n — oc, and thus /(t m ,p(t)) = 0. Thus (2.6) y = ^ 6 ^ is a branch of the curve / = 0. This expansion is called a Puiseux series (when ro = mult(/)), in honor of Puiseux, who introduced this theory into algebraic geometry. Remark 2.2. Our proof of Theorem 2.1 is not valid in positive character- istic, since we cannot conclude (2.4). Theorem 2.1 is in fact false over fields of positive characteristic. See Exercise 2.4 at the end of this section. Suppose that / G if[[x,y]] is irreducible, and that we have found a solution y = p(x™) to f(x,y) = 0. We may suppose that m is the smallest natural number for which it is possible to write such a series, y — p(x™) divides / in R\ — if[[xm,i/]], Let u b e a primitive m-th root of unity in K. Since / is invariant under the iiT-algebra automorphism fi of R\ determined by #m — u xm and y —• y, it follows that y — p{u)^x^)) \ f in R\ for all j , and thus y — p(u)ix™) is a solution to /(#,y) = 0 for all j . These solutions are distinct for 0 j m — 1, by our choice of ra. The series m—1 9= n (?/-P(^J^)) is invariant under (/, so y E if[[x,y]] and y | / in iiT[[x,y]], the ring of invariants of R\ under the action of the group Z m generated by 0. Since / is irreducible, ( ra-l 3=0 where u is a unit in if[[x,y]]. Remark 2.3. Some letters of Newton developing this idea are translated (from Latin) in [18]. After we have defined non-singularity, we will return to this algorithm in (2.7) of Section 2.5, to see that we have actually constructed a resolution of singularities of a plane curve singularity. Exercise 2.4. 1. Construct a Puiseux series solution to f{x, y) = yA - 2x3y2 - 4x5y + x6 - x7 = 0 over the complex numbers.