diff --git a/src/LocalField/LocalField.jl b/src/LocalField/LocalField.jl
index 335626b38a..dc86d9f795 100644
--- a/src/LocalField/LocalField.jl
+++ b/src/LocalField/LocalField.jl
@@ -47,8 +47,8 @@ elem_type(::Type{LocalField{S, T}}) where {S <: FieldElem, T <: LocalFieldParame
 #
 ################################################################################
 
-is_domain_type(::Type{S}) where S <: LocalField = true
-is_exact_type(::Type{S}) where S <: LocalField = false
+is_domain_type(::Type{<: LocalFieldElem}) = true
+is_exact_type(::Type{<: LocalFieldElem}) = false
 isfinite(K::LocalField) = isfinite(base_field(K))
 
 ################################################################################
diff --git a/src/Misc/RelFiniteField.jl b/src/Misc/RelFiniteField.jl
index 3d5eee6d01..b6b16b6a34 100644
--- a/src/Misc/RelFiniteField.jl
+++ b/src/Misc/RelFiniteField.jl
@@ -141,6 +141,8 @@ iszero(x::RelFinFieldElem) = iszero(x.data)
 isone(x::RelFinFieldElem) = isone(x.data)
 is_unit(x::RelFinFieldElem) = !iszero(x)
 
+hash(x::RelFinFieldElem, h::UInt) = hash(x.data, h)
+
 ==(x::RelFinFieldElem{S, T}, y::RelFinFieldElem{S, T}) where {S, T} = x.data == y.data
 
 coeff(a::RelFinFieldElem, i::Int) = coeff(a.data, i)
diff --git a/src/NumField/Embedded.jl b/src/NumField/Embedded.jl
index 9612a79154..5b5ee17116 100644
--- a/src/NumField/Embedded.jl
+++ b/src/NumField/Embedded.jl
@@ -65,6 +65,8 @@ base_ring_type(::Type{<:EmbeddedNumField}) = QQField
 
 data(x::EmbeddedNumFieldElem) = x.element
 
+characteristic(::EmbeddedNumField) = 0
+
 function embedded_field(K::SimpleNumField, i::NumFieldEmb)
   @assert number_field(i) === K
   E = EmbeddedNumField(K, i)