Eisenstein’s criterion gives us a sufficient criterion for a property of polynomials with integer coefficients that is difficult to capture in general: irreducibility, meaning the impossibility of factoring it with rational coefficients. We apply this criterion to a prime $p$’s cyclotomic polynomial by applying a shift, which introduces binomial coefficients from the $p^{\text{th}}$ row of Pascal’s triangle. These coefficients are known to be divisible by $p$, which allows us to invoke Eisenstein.