Exercise 21, § 3

Show that in $\mathbf{Z}[X]$ the following polynomials are irreducible: $X^n - a$, where one of the prime factors of $a$ has exponent 1 ; $X^{2^k} + 1$, (replace $X$ by $X + 1$); $x^4 + 3x^3 + 3x^2 - 5$; $5X^4 - 6X^3 - aX^2 - 4X + 2$. (Use Exercise 20.)