For indirect proof, suppose that there are only finitely many irreducible polynomials f₁, f₂, ^{. . .} , f_n, with f_n having the highest possible degree that an irreducible polynomial can have. Then the polynomial f₁f₂ ^{. . .} f_n + 1 has degree even higher than that of f_n and so must be reducible. Hence, it is divisible by some f_i, which is impossible.