diff --git a/src/NumField/Elem.jl b/src/NumField/Elem.jl
index 5e0d6df889..1321a4e586 100644
--- a/src/NumField/Elem.jl
+++ b/src/NumField/Elem.jl
@@ -447,16 +447,34 @@ coefficientwise.
 """
 function norm(f::PolyRingElem{<: NumFieldElem})
   K = base_ring(f)
-  P = polynomial_to_power_sums(f, degree(f)*degree(K))
-  PQ = elem_type(base_field(K))[tr(x) for x in P]
-  return power_sums_to_polynomial(PQ)
+  n, ff = remove(f, gen(parent(f)))
+  if degree(ff) > 0
+    P = polynomial_to_power_sums(ff, degree(ff)*degree(K))
+    PQ = elem_type(base_field(K))[tr(x) for x in P]
+    N = power_sums_to_polynomial(PQ)
+  else
+    k = base_field(K)
+    kt, = polynomial_ring(k, :t; cached = false)
+    N = kt(norm(constant_coefficient(ff)))
+  end
+  (n > 0) && (N = shift_left(N, n*degree(K)))
+  return N
 end
 
 function norm(f::PolyRingElem{<:NumFieldElem}, k::NumField)
   K = base_ring(f)
-  P = polynomial_to_power_sums(f, degree(f)*degree(K))
-  PQ = elem_type(base_field(K))[tr(x, k) for x in P]
-  return power_sums_to_polynomial(PQ)
+  n, ff = remove(f, gen(parent(f)))
+  if degree(ff) > 0
+    P = polynomial_to_power_sums(ff, degree(ff)*degree(K))
+    PQ = elem_type(base_field(K))[tr(x, k) for x in P]
+    N = power_sums_to_polynomial(PQ)
+  else
+    k = base_field(K)
+    kt, = polynomial_ring(k, :t; cached = false)
+    N = kt(norm(constant_coefficient(ff), k))
+  end
+  (n > 0) && (N = shift_left(N, n*degree(K)))
+  return N
 end
 
 norm(a::QQPolyRingElem) = a
@@ -686,7 +704,7 @@ end
 
 absolute_minpoly(a::AbsSimpleNumFieldElem) = minpoly(a)
 
-absolute_minpoly(a::AbsNonSimpleNumField) = minpoly(a)
+absolute_minpoly(a::AbsNonSimpleNumFieldElem) = minpoly(a)
 
 absolute_minpoly(a::T) where T <: Union{RelNonSimpleNumFieldElem, RelSimpleNumFieldElem} = minpoly(a, QQ)
 
diff --git a/test/NumField/Elem.jl b/test/NumField/Elem.jl
index 147a8f2ebb..95a36572dc 100644
--- a/test/NumField/Elem.jl
+++ b/test/NumField/Elem.jl
@@ -287,3 +287,35 @@ let # issue #2079, Hanselmann
   f = L(b) * u^2 + L(c)
   @test length(roots(f)) == 2
 end
+
+let
+  Qx, x = QQ["x"]
+  K, a = number_field([x^2 - 2, x^2 - 3])
+  @test absolute_minpoly(a[1])(x) == x^2 - 2
+end
+
+let # norm of polynomials
+  QQ, = rationals_as_number_field()
+  Qx, xQ = QQ["x"]
+  F, _ = number_field(xQ^2-2)
+  Fx, xF = F["x"]
+  d = degree(F)
+  @test norm(xF)(xQ) == xQ^(d)
+  for i in 1:11
+    @test norm(xF^i + xF^(i+1))(xQ) == xQ^(i*d) * norm(xF+1)(xQ)
+  end
+end
+
+let
+  QQ, = rationals_as_number_field()
+  Qx, xQ = QQ["x"]
+  F, a = number_field(xQ^2 - 2)
+  Fx, xF = F["x"]
+  L, z = number_field(xF^2 - 3)
+  Lt, t = L["t"]
+  d = degree(F)
+  @test norm(xF, F)(xQ) == xQ^(d)
+  for i in 1:11
+    @test norm(t^i + t^(i+1), F)(xF) == xF^(i*d) * norm(t + 1, F)(xF)
+  end
+end