Towards non-nice relative number fields (#1209)

src/NumField/Elem.jl

@@ -658,3 +658,18 @@ absolute_minpoly(a::nf_elem) = minpoly(a)
 absolute_minpoly(a::NfAbsNS) = minpoly(a)
 
 absolute_minpoly(a::T) where T <: Union{NfRelNSElem, NfRelElem} = minpoly(a, FlintQQ)
+
+################################################################################
+#
+#  Integral multiplicator
+#
+################################################################################
+
+function _integral_multiplicator(a::Union{PolyElem, MPolyElem})
+  return lcm(ZZRingElem[_integral_multiplicator(c) for c in coefficients(a)])
+end
+
+function _integral_multiplicator(a::NumFieldElem)
+  f = minpoly(a)
+  return _integral_multiplicator(f)
+end

src/NumField/NfAbs/Elem.jl

@@ -950,3 +950,13 @@ function complex_conjugate(a::nf_elem)
   end
   error("Not implemented yet: element must be in an at most quadratic field")
 end
+
+################################################################################
+#
+#  Integral multiplicator
+#
+################################################################################
+
+_integral_multiplicator(a::nf_elem) = denominator(minpoly(a))
+
+_integral_multiplicator(a::QQPolyRingElem) = denominator(a)

src/NumField/NfRel/NfRel.jl

@@ -922,3 +922,13 @@ function cyclotomic_field_as_cm_extension(n::Int; cached::Bool = true)
   return E, b
 end
 
+#################################################################################
+##
+##  Integral multiple
+##
+#################################################################################
+#
+#function _integral_multiplicator(a::NfRelElem)
+#  f = minpoly(a)
+#  return lcm(ZZRingElem[_integral_multiplicator(c) for c in coefficients(f)])
+#end

src/NumField/NfRel/NfRelNS.jl

@@ -822,3 +822,14 @@ function is_rational(a::NfRelNSElem)
   d = data(a)
   return is_constant(d) && is_rational(constant_coefficient(d))
 end
+#
+#################################################################################
+##
+##  Integral multiplicator
+##
+#################################################################################
+#
+#function _integral_multiplicator(a::NfRelNSElem)
+#  f = minpoly(a)
+#  return lcm(ZZRingElem[_integral_multiplicator(c) for c in coefficients(f)])
+#end

src/NumFieldOrd/NfOrd/Ideal/Prime.jl

@@ -904,6 +904,9 @@ end
 
 function _prefactorization(I::NfOrdIdl)
   @assert has_2_elem(I)
+  if !is_defining_polynomial_nice(nf(I))
+    return __prefactorization(I)
+  end
   n = I.gen_one
   if has_minimum(I)
     n = minimum(I)
@@ -919,6 +922,10 @@ function _prefactorization(I::NfOrdIdl)
 end
 
 function _prefactorization(I::NfAbsOrdIdl)
+  return __prefactorization(I)
+end
+
+function __prefactorization(I::NfAbsOrdIdl)
   return coprime_base(ZZRingElem[I.gen_one, norm(I), minimum(I)])
 end
 

src/NumFieldOrd/NfOrd/NfOrd.jl

@@ -882,17 +882,24 @@ function any_order(K::AnticNumberField)
 end
 
 function any_order(K::NfAbsNS)
-  normalized_gens = Vector{NfAbsNSElem}(undef, degree(K))
+  normalized_gens = Vector{NfAbsNSElem}(undef, ngens(K))
   g = gens(K)
   for i in 1:ngens(K)
     f = denominator(K.pol[i]) * K.pol[i]
+    @show f
+    @show isone(coeff(f, 1))
+    @show coeff(f, 1)
+    @show typeof(f)
+    @show g[i]
     if isone(coeff(f, 1))
       normalized_gens[i] = g[i]
     else
       normalized_gens[i] = coeff(f, 1) * g[i]
     end
   end
 
+  @show normalized_gens
+
   b = Vector{NfAbsNSElem}(undef, degree(K))
   ind = 1
   it = cartesian_product_iterator([1:degrees(K)[i] for i in 1:ngens(K)], inplace = true)

src/NumFieldOrd/NfOrd/ResidueField.jl

@@ -197,7 +197,7 @@ function residue_field(O::NfOrd, P::NfOrdIdl, check::Bool = true)
   if check
     !is_prime(P) && error("Ideal must be prime")
   end
-  if !is_maximal_known(O) || !is_maximal(O)
+  if !is_maximal_known(O) || !is_maximal(O) || !is_defining_polynomial_nice(nf(O))
     return _residue_field_generic_fq_default(O, P)
   end
   if !is_index_divisor(O, minimum(P)) && has_2_elem(P)

src/NumFieldOrd/NfRelOrd/NfRelOrd.jl

@@ -506,7 +506,52 @@ function EquationOrder(L::NumField)
   return O
 end
 
+function any_order(K::NfRel)
+  f = defining_polynomial(K)
+  de = _integral_multiplicator(f)
+  g = f * de
+  if is_monic(g)
+    return equation_order(K)
+  else
+    d = degree(g)
+    M = zero_matrix(base_field(K), d, d)
+    M[1, 1] = 1
+    for i in 2:d
+      for j in i:-1:2
+        M[i, j] = coeff(g, d - (i - j))
+      end
+    end
+    B = pseudo_hnf(pseudo_matrix(M), :lowerleft)
+    return Order(K, B)
+  end
+end
+
+function any_order(K::NfRelNS)
+  normalized_gens = Vector{elem_type(K)}(undef, ngens(K))
+  g = gens(K)
+  pols = defining_polynomials(K)
+  for i in 1:ngens(K)
+    f = _integral_multiplicator(K.pol[i]) * K.pol[i]
+    if isone(coeff(f, 1))
+      normalized_gens[i] = g[i]
+    else
+      normalized_gens[i] = coeff(f, 1) * g[i]
+    end
+  end
+
+  b = Vector{elem_type(K)}(undef, degree(K))
+  ind = 1
+  it = cartesian_product_iterator([1:degrees(K)[i] for i in 1:ngens(K)], inplace = true)
+  for i in it
+    b[ind] = prod(normalized_gens[j]^(i[j] - 1) for j=1:length(i))
+    ind += 1
+  end
+  return Order(K, basis_matrix(b))
+end
+
 function EquationOrder(L::NfRel{nf_elem})
+  a = gen(L)
+  @req is_integral(a) "Generator of must be integral"
   M = identity_matrix(base_field(L), degree(L))
   PM = pseudo_matrix(M)
   O = Order(L, PM)

test/NfRel/NfRelOrd.jl

@@ -212,3 +212,19 @@ K, a = quadratic_field(5)
 Kt, t = K["t"]
 L, b = number_field(polynomial(K, [-2, 0, 0, 1]), "b");
 @test_throws Hecke.NotImplemented extend(equation_order(L), [b])
+
+# Towards non-nice equations
+
+begin
+  Qx, x = QQ["x"]
+  K, a = number_field(x - 1)
+  Kt, t = K["t"]
+  L, b = number_field(t^2 + 1//2)
+  c = Hecke._integral_multiplicator(b)
+  @test is_integral(c * b)
+  O = any_order(L)
+  @test nf(O) === L
+  L, b = number_field([t^2 + 1//2])
+  O = any_order(L)
+  @test nf(O) === L
+end