Show any straight line is irreducible

Question

Show that any straight line in $\mathbb{F}^{n}$ is irreducible, where F is an infinite field.

I know V($ax+b$) would be a variety that represents any straight line and then V is irreducible if I(V) is prime but I'm not too sure where to go from here.

Answer 1

Use that lines can be parameterised.

Cox Little and O'Shea: Ideals, Varieties and Algorithms

Proposition 5. If $k$ is an infinite field and $V \subset k^n$ is a variety defined parametrically $$x_1 = f_1(t_1,\ldots,t_m), $$ $$\vdots$$ $$x_n = f_n(t_1,\ldots,t_m),$$ where $f_1, . . . , f_n$ are polynomials in $k[t_1, \ldots , t_m ]$, then $V$ is irreducible.