The letter (Q) is the symbol that is used to represent rational numbers. Rational numbers are sometimes called fractions. They are numbers that can be written as the quotient of two integers. They have decimal representations that either terminate or do not terminate but contain a repeating block of digits.

for some unit u ∈ R. Since Q(R) is a field, so Q(R)[x] is a UFD. Also the irreducible polynomials p1(x),p2(x),··· ,pm(x),q1(x),q2(x),··· ,qn(x) ∈ R[x] are primitive, which implies that they are also irreducible in Q(R)[x].

A rational number like 1927 is a quotient of two integers. Collectively, the rational numbers are denoted Q, where the letter Q is the first letter of the word “quotient”. The rational numbers have all four arithmetic operations—addition, subtraction, multiplication, and division—so they form what is called a field.

Definition. Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero element x of R can be written as a product of a unit u and zero or more irreducible elements p_i of R: x = u p₁ p₂ ⋅⋅⋅ p_n with n ≥ 0.