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§4. Rational Cubics and Mordell’s Theorem 11

with a line, but, instead, we move between rational points within the cubic to give
the cubic an algebraic structure. This is called the chord-tangent law of composition.
The intersection result needed for a line and a cubic, which is related to (2.2), is
the following.
(4.1) Remark. If two of the three intersection points of a rational cubic with a ra-
tional line are rational points, then the third point is rational.
To see this, we use the equation aw + bx + cy = 0 of the line to eliminate one
variable in the third-order equation F(w, x, y) = 0 of the cubic. For intersections
off the line at infinity, given by w = 0, one comes up with a cubic equation in the
x-coordinate or in the y-coordinate of the intersection points. Thus the intersection
points will be rational if and only if the roots of the cubic equation are rational.
Thus (4.1) reduces to the algebraic statement: if a cubic polynomial with rational
coefficients has two rational roots, then the third root is rational.

(4.2) Definition. An irreducible cubic is one which cannot be factored over the com-
plex numbers. A point O on a irreducible cubic C is called a singular point provided
each line through O intersects C at, at most, one other point. An irreducible cubic
without a singular point is called a nonsingular cubic curve, and one with singular
points is called a singular cubic.
The description of rational points on rational cubics, which are either reducible
or singular cubics follows very much the ideas used for conics. First we consider a
2
2
cubic with a singular rational point O. A typical example is given by y = x (x +a)
and O = (0, 0), the origin.

Since O is a singular point, each rational line L through O cuts the cubic at a second
point P,and P is rational because its x-coordinate is the solution of a cubic equation
in x or in y with a double rational root correspondingto the x-or y-coordinate of O.
Thus, as with conics, we can project the singular cubic onto any fixed rational line
M in such a way that rational points on the cubic correspond to rational points on the
line M.
Next we consider a nonsingular cubic. A line meets these cubics in three points in
¨
general, and if we have one rational point, one cannot project the cubic in the naive