feat: add roots/factor for embedded field polys (#1347)

Just foward the methods to the underlying number field.

Project.toml

@@ -1,6 +1,6 @@
 name = "Hecke"
 uuid = "3e1990a7-5d81-5526-99ce-9ba3ff248f21"
-version = "0.24.0"
+version = "0.24.1"
 
 [deps]
 AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"

src/NumField/Embedded.jl

@@ -287,3 +287,25 @@ end
 function is_rational(x::EmbeddedElem)
   return is_rational(data(x))
 end
+
+################################################################################
+#
+#  Factorization and roots
+#
+################################################################################
+
+function roots(f::PolyRingElem{<:EmbeddedElem})
+  E = base_ring(parent(f))
+  K = number_field(E)
+  return E.(roots(map_coefficients(data, f, cached = false)))
+end
+
+function factor(f::PolyRingElem{<:EmbeddedElem})
+  Ex = parent(f)
+  E = base_ring(parent(f))
+  K = number_field(E)
+  fa = factor(map_coefficients(data, f, cached = false))
+  return Fac(Ex(E(constant_coefficient(unit(fa)))),
+             Dict{typeof(f), Int}(
+               (map_coefficients(E, g, parent = Ex), e) for (g, e) in fa))
+end

test/NumField/Embedded.jl

@@ -90,5 +90,19 @@ test_elem(E::Hecke.EmbeddedField) = E(rand(number_field(E), -10:10))
   @test @inferred is_rational(a^0)
   @test !is_rational(a)
   @test (@inferred QQ(2*a^0)) == 2 * one(QQ)
+
+  # roots and factor
+  begin
+    Qx, x = QQ["x"]
+    K, _a = number_field(x^2 - 2, "a")
+    i = Hecke.real_embedding(K, 1.41)
+    E, a = Hecke.embedded_field(K, i)
+    Et, t = E["t"]
+    @test issetequal(roots(t^2 - 2), [a, -a])
+    fa = factor(t^2 - 2)
+    @test unit(fa) * prod(g^e for (g, e) in fa) == t^2 - 2
+    fa = factor(t^2 - a)
+    @test unit(fa) * prod(g^e for (g, e) in fa) == t^2 - a
+  end
 end