Fix local fundamental class

Project.toml

@@ -1,6 +1,6 @@
 name = "Hecke"
 uuid = "3e1990a7-5d81-5526-99ce-9ba3ff248f21"
-version = "0.22.6"
+version = "0.22.7"
 
 [deps]
 AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"

src/LocalField/neq.jl

@@ -448,7 +448,14 @@ end
 absolute_representation_matrix(a::fqPolyRepField) = representation_matrix(a)
 absolute_representation_matrix(a::fpFieldElem) = matrix(parent(a), 1, 1, [a])
 
-function absolute_representation_matrix(a::RelFinFieldElem)
+function absolute_frobenius_matrix(a::FqField, d::Int = 1)
+  b = absolute_basis(a)
+  b = [absolute_frobenius(x, d) for x = b]
+  return matrix([absolute_coordinates(x) for x = b])
+end
+
+
+function absolute_representation_matrix(a::Union{RelFinFieldElem, FqFieldElem})
   b = a .* absolute_basis(parent(a))
   return matrix([absolute_coordinates(x) for x = b])
 end

src/Misc/RelFiniteField.jl

@@ -497,6 +497,15 @@ function absolute_basis(F::T) where T <: FinField
   return powers(gen(F), degree(F)-1)
 end
 
+function absolute_basis(F::FqField)
+  return powers(Nemo._gen(F), absolute_degree(F) - 1)
+end
+
+function absolute_coordinates(x::FqFieldElem)
+  F = prime_field(parent(x))
+  return FqFieldElem[F(Nemo._coeff(x, i)) for i in 0:Nemo._degree(parent(x))-1]
+end
+
 function absolute_coordinates(x::FinFieldElem)
   F = parent(x)
   Fp = prime_field(F)

test/Misc/RelFinField.jl

@@ -94,6 +94,29 @@
         end
       end
     end
+
+    F = Hecke.Nemo._GF(3^3, cached = false)
+    x = polynomial_ring(F, "x", cached = false)[2]
+    K, gK = Hecke.Nemo._residue_field(x^2+1, "a")
+    Kt, t = K["t"]
+    L, gL = Hecke.Nemo._residue_field(t^5+t^4+t^2+1, "b")
+    B = absolute_basis(L)
+    for i = 1:length(B)
+      v = absolute_coordinates(B[i])
+      for j = 1:length(v)
+        if i == j
+          @test isone(v[j])
+        else
+          @test iszero(v[j])
+        end
+      end
+    end
+
+    for i in 1:100
+      z = rand(L)
+      @test dot(absolute_coordinates(z), absolute_basis(L)) == z
+    end
+
   end
 
   @testset "Polynomials" begin