consider only the branches of the curve that pass through the origin. Without loss of generality, we may suppose that is coprime with the greatest common divisor of the such that (otherwise, one could choose a smaller common denominator for the exponents). Let be a primitive th root of unity. If the above Puiseux series occurs in the factorization of , then the series occur also in the factorization (a consequence of Galois theory). These series are said conjugate, and are considered as a single branch of the curve, of ramification index . In the case of a real curve, that is a curve defined by a polynomial with real coefficients, three cases may occur. If none has real coefficients, then one has a non-real branch. If some has real coefficients, then one may choose it as . If is odd, then every real value of provides a real value of , and one has a real branch that looks regular, although it is singular if . If is even, then and have real values, but only for . In this case, the real branch looks as a cusp (or is a cusp, depending on the definition of a cusp that is used). For example, the ordinary cusp has only one branch. If it is defined by the equation then the factorization is the ramification index is 2, and the two factors are real and define each a half branch. If the cusp is rotated, it equation becomes and the factorization is with (the coefficient has not been simplified to for showing how the above definition of is specialized). Here the ramification index is 3, and only one factor is real; this shows that, in the first case, the two factors must be considered as defining the same branch. Non-plane algebraic curves An algebraic curve is an algebraic variety of dimension one. This implies that an affine curve in an affine space of dimension n is defined by, at least, n − 1 polynomials in n variables. To define a curve, these polynomials must generate a prime ideal of Krull dimension 1. This condition is not easy to test in practice. Therefore, the following way to represent non-plane curves may be preferred. Let be n polynomials in two variables x1 and x2 such that f is irreducible. The points in the affine space of dimension n such whose coordinates satisfy