In this article, we compile the work done by various mathematicians on the topic of the fixed divisor of a polynomial. This article explains most of the results concisely and is intended to be an exhaustive survey. We present the results on fixed divisors in various algebraic settings as well as the applications of fixed divisors to various algebraic and number theoretic problems. The work is presented in an orderly fashion so as to start from the simplest case of Z, progressively leading up to the case of Dedekind domains. We also ask a few open questions according to their context, which may give impetus to the reader to work further in this direction. We describe various bounds for fixed divisors as well as the connection of fixed divisors with different notions in the ring of integer-valued polynomials. Finally, we suggest how the generalization of the ring of integer-valued polynomials in the case of the ring of n × n matrices over Z (or Dedekind domain) could lead to the generalization of fixed divisors in that setting.keywords Fixed divisors, Generalized factorials, Generalized factorials in several variables, Common factor of indices, Factoring of prime ideals, Integer valued polynomials NotationsWe fix the notations for the whole paper. R = Integral Domain K = Field of fractions of R N (I) = Cardinality of R/I (Norm of an ideal I ⊆ R) W = {0, 1, 2, 3, . . .} A[x] = Ring of polynomials in n variables (= A[x 1 , . . . , x n ]) with coefficients in the ring A S = Arbitrary (or given) subset of R n such that no non-zero polynomial in K[x] maps it to zero S = S in case when n = 1 Int(S, R) = Polynomials in K[x] mapping S back to R ν k (S) = Bhargava's (generalized) factorial of index k k! S = k th generalized factorial in several variables M m (S) = Set of all m × m matrices with entries in S p = positive prime number Z p = p-adic integers ord p (n) = p-adic ordinal (valuation) of n ∈ Z.

The ring of integer-valued polynomials over a given subset S of Z (or Int(S, Z)) is defined as the set of polynomials in Q[x] which maps S to Z. In factorization theory, it is crucial to check the irreducibility of a polynomial. In this article, we make Bhargava factorials our main tool to check the irreducibility of a given polynomial f ∈ Int(S, Z)). We also generalize our results to arbitrary subsets of a Dedekind domain.

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