There are efficient algorithms for testing polynomial irreducibility and factoring polynomials over finite field. Hilbert's irreducibility theorem has numerous applications in number theory and algebra. In other words, atomicity means indivisibility and irreducibility. In this section, one does not need irreducibility of algebraic sets nor primality of ideals. Gauss's lemma can also be used to show Eisenstein's irreducibility criterion. Computational irreducibility explains observed limitations of existing mainstream science. George Mackey established a criterion to verify the irreducibility of induced representations. The irreducibility of this algebraic set needs a proof. The notions of irreducibility in algebra and manifold theory are related. Eisenstein's criterion can also be used in some cases to determine irreducibility. In cases of computational irreducibility, only observation and experiment can be used. Computational irreducibility may also provide a scientifically-based resolution for free will. But by irreducibility of the Pauli matrices, the representation cannot be further reduced. The projectionist recognizes the irreducibility of human identity to laws of nature. Janiszewski, however, focused on the irreducibility of these continua.