 Familiar theorems of number theory emerge in algebraic form for algebras. An algebra \(A\) is called directly irreducible or simple when its lattice of congruences is the two-element lattice consisting of \(A\) and the one-element algebra, paralleling the notion of prime number \(p\) as a number whose lattice of divisors has two elements \(p\) and 1. However the counterpart of the fundamental theorem of arithmetic, that every positive integer factors uniquely as a product of primes, requires a more delicate kind of product than direct product. Birkhoff’s notion of subdirect product enabled him to prove the Subdirect Representation Theorem, that every algebra arises as the subdirect product of its subdirectly irreducible quotients. Whereas there are many subdirectly irreducible groups, the only subdirectly irreducible Boolean algebra is the initial or two-element one, while the subdirectly irreducible rings satisfying \(x^n = x\) for some \(n \gt 1\) are exactly the finite fields.