diff --git a/docs/src/division.md b/docs/src/division.md
index fe11e597..a20a6b1d 100644
--- a/docs/src/division.md
+++ b/docs/src/division.md
@@ -14,6 +14,10 @@ div_multiple
 
 Note that the coefficients of the polynomials need to be a field for `div`,
 `rem` and `divrem` to work.
+If the coefficient type is not a field, it is promoted to a field using [`promote_to_field`](@ref).
+```@docs
+promote_to_field
+```
 Alternatively, [`pseudo_rem`](@ref) or [`pseudo_divrem`](@ref) can be used
 instead as they do not require the coefficient type to be a field.
 ```@docs
diff --git a/src/division.jl b/src/division.jl
index 7ca299c4..3541c332 100644
--- a/src/division.jl
+++ b/src/division.jl
@@ -64,8 +64,25 @@ struct Field end
 struct UniqueFactorizationDomain end
 const UFD = UniqueFactorizationDomain
 
+"""
+    promote_to_field(::Type{T})
+
+Promote the type `T` to a field. For instance, `promote_to_field(T)` returns
+`Rational{T}` if `T` is an integer and `promote_to_field(T)` returns `RationalPoly{T}`
+if `T` is a polynomial.
+"""
+function promote_to_field end
+
+function promote_to_field(::Type{T}) where {T<:Integer}
+    return Rational{T}
+end
+function promote_to_field(::Type{T}) where {T<:_APL}
+    return RationalPoly{T,T}
+end
+promote_to_field(::Type{T}) where {T} = T
+
 algebraic_structure(::Type{<:Integer}) = UFD()
-algebraic_structure(::Type{<:AbstractPolynomialLike}) = UFD()
+algebraic_structure(::Type{<:_APL}) = UFD()
 # `Rational`, `AbstractFloat`, JuMP expressions, etc... are fields
 algebraic_structure(::Type) = Field()
 _field_absorb(::UFD, ::UFD) = UFD()
@@ -430,7 +447,7 @@ function MA.promote_operation(
     ::Type{P},
     ::Type{Q},
 ) where {T,S,P<:_APL{T},Q<:_APL{S}}
-    U = MA.promote_operation(/, T, S)
+    U = MA.promote_operation(/, promote_to_field(T), promote_to_field(S))
     # `promote_type(P, Q)` is needed for TypedPolynomials in case they use different variables
     return polynomial_type(promote_type(P, Q), MA.promote_operation(-, U, U))
 end
diff --git a/test/division.jl b/test/division.jl
index 36b995ab..cce74b1e 100644
--- a/test/division.jl
+++ b/test/division.jl
@@ -54,8 +54,14 @@ end
 
 function divrem_test()
     Mod.@polyvar x y
-    @test (@inferred div(x * y^2 + 1, x * y + 1)) == y
-    @test (@inferred rem(x * y^2 + 1, x * y + 1)) == -y + 1
+    p = x * y^2 + 1
+    q = x * y + 1
+    @test typeof(div(p, q)) == MA.promote_operation(div, typeof(p), typeof(q))
+    @test typeof(rem(p, q)) == MA.promote_operation(rem, typeof(p), typeof(q))
+    @test coefficient_type(div(p, q)) == Rational{Int}
+    @test coefficient_type(rem(p, q)) == Rational{Int}
+    @test (@inferred div(p, q)) == y
+    @test (@inferred rem(p, q)) == -y + 1
     @test (@inferred div(x * y^2 + x, y)) == x * y
     @test (@inferred rem(x * y^2 + x, y)) == x
     @test (@inferred rem(x^4 + x^3 + (1 + 1e-10) * x^2 + 1, x^2 + x + 1)) == 1