Every nonempty K-variety in Fⁿ may be written uniquely as a finite union V₁ U V₂ U · · · U V_k of affine K-varieties in F_n such that V_j ⊄ V_i for i ≠ j and each V_i is irreducible (Exercise 8).

Data from exercise 8

A K-variety V in Fⁿ is irreducible provided that whenever V = W₁ U W₂ with each W_i a K-variety in F_n, either V = W₁ or V = W₂.

(a) Prove that V is irreducible if and only if J(V) is a prime ideal in K[x₁, ... , x_n].

(b) Let F = C and S = { x1² - 2x₂² l. Then V(S) is irreducible as a Q-variety but not as an R-variety.