Change residue_field for local fields

Project.toml

@@ -37,7 +37,7 @@ LazyArtifacts = "1.6"
 Libdl = "1.6"
 LinearAlgebra = "1.6"
 Markdown = "1.6"
-Nemo = "^0.37.0"
+Nemo = "^0.37.4"
 Pkg = "1.6"
 Polymake = "0.10, 0.11"
 Printf = "1.6"

src/LocalField/Completions.jl

@@ -152,7 +152,8 @@ function completion(K::AnticNumberField, P::NfOrdIdl, precision::Int = 64)
   Qq, gQq = QadicField(minimum(P), f, prec_padics, cached = false)
   Qqx, gQqx = polynomial_ring(Qq, "x")
   q, mq = residue_field(Qq)
-  F, mF = ResidueFieldSmall(OK, P)
+  #F, mF = ResidueFieldSmall(OK, P)
+  F, mF = residue_field(OK, P)
   mp = find_morphism(q, F)
   g = gen(q)
   gq_in_K = (mF\(mp(g))).elem_in_nf
@@ -444,7 +445,7 @@ function unramified_completion(K::AnticNumberField, P::NfOrdIdl, precision::Int
   Qp = PadicField(p, precision)
   Zp = maximal_order(Qp)
   q, mq = residue_field(Qq)
-  F, mF = ResidueFieldSmall(OK, P)
+  F, mF = residue_field(OK, P)
   mp = find_morphism(q, F)
   g = gen(q)
   gq_in_K = (mF\(mp(g))).elem_in_nf

src/LocalField/Conjugates.jl

@@ -558,21 +558,21 @@ function completion(K::AnticNumberField, ca::qadic)
   while length(pa) < d
     push!(pa, pa[end]*pa[2])
   end
-  m = matrix(Native.GF(p), d, d, [coeff(pa[i], j-1) for j=1:d for i=1:d])
-  o = matrix(Native.GF(p), d, 1, [coeff(gen(R), j-1) for j=1:d])
+  m = matrix(Nemo._GF(p), d, d, [lift(ZZ, coeff(pa[i], j-1)) for j=1:d for i=1:d])
+  o = matrix(Nemo._GF(p), d, 1, [lift(ZZ, coeff(gen(R), j-1)) for j=1:d])
   s = solve(m, o)
   @hassert :qAdic 1 m*s == o
   a = K()
   for i=1:d
-    _num_setcoeff!(a, i-1, lift(s[i,1]))
+    _num_setcoeff!(a, i-1, lift(ZZ, s[i, 1]))
   end
   f = defining_polynomial(parent(ca), FlintZZ)
   fso = inv(derivative(f)(gen(R)))
-  o = matrix(Native.GF(p), d, 1, [coeff(fso, j-1) for j=1:d])
+  o = matrix(Nemo._GF(p), d, 1, [lift(ZZ, coeff(fso, j-1)) for j=1:d])
   s = solve(m, o)
   b = K()
   for i=1:d
-    _num_setcoeff!(b, i-1, lift(s[i,1]))
+    _num_setcoeff!(b, i-1, lift(ZZ, s[i,1]))
   end
 
   #TODO: don't use f, use the factors i the HenselCtx

src/LocalField/LocalField.jl

@@ -465,7 +465,7 @@ function residue_field(K::LocalField{S, UnramifiedLocalField}) where {S <: Field
    Fpt = polynomial_ring(ks, cached = false)[1]
    g = defining_polynomial(K)
    f = Fpt([ks(mks(coeff(g, i))) for i=0:degree(K)])
-   kk = Native.finite_field(f)[1]
+   kk, = Nemo._residue_field(f)
    bas = basis(K)
    u = gen(kk)
    function proj(a::Hecke.LocalFieldElem)
@@ -494,9 +494,10 @@ end
 
  function unramified_extension(L::Union{FlintPadicField, FlintQadicField, LocalField}, n::Int)
    R, mR = residue_field(L)
-   f = polynomial(R, push!([rand(R) for i = 0:n-1], one(R)))
+   Rt, t = polynomial_ring(R, "t", cached = false)
+   f = Rt(push!([rand(R) for i = 0:n-1], one(R)))
    while !is_irreducible(f)
-     f = polynomial(R, push!([rand(R) for i = 0:n-1], one(R)))
+     f = Rt(push!([rand(R) for i = 0:n-1], one(R)))
    end
    f_L = polynomial(L, [mR\(coeff(f, i)) for i = 0:degree(f)])
    return unramified_extension(f_L)

src/LocalField/Poly.jl

@@ -78,6 +78,10 @@ function lift(a::T, K::PadicField) where T <: Union{Nemo.zzModRingElem, Nemo.ZZM
   return Hecke.lift(a) + O(K, p^v)
 end
 
+function lift(a::FqFieldElem, K::PadicField)
+  return Hecke.lift(ZZ, a) + O(K, prime(K))
+end
+
 function lift(a::FinFieldElem, K::LocalField)
   k, mk = residue_field(K)
   @assert k === parent(a)
@@ -883,6 +887,8 @@ mutable struct HenselCtxdr{S}
 #    @assert sum(map(degree, lfp)) == degree(f)
     Q = base_ring(f)
     Qx = parent(f)
+    @assert residue_field(Q)[1] === coefficient_ring(lfp[1])
+    k, Qtok = residue_field(Q)
     i = 1
     la = Vector{typeof(f)}()
     n = length(lfp)
@@ -891,12 +897,12 @@ mutable struct HenselCtxdr{S}
       f2 = lfp[i+1]
       g, a, b = gcdx(f1, f2)
       @assert isone(g)
-      push!(la, map_coefficients(x -> setprecision(lift(x, Q), 1), a, parent = Qx))
-      push!(la, map_coefficients(x -> setprecision(lift(x, Q), 1), b, parent = Qx))
+      push!(la, map_coefficients(x -> setprecision(Qtok\x, 1), a, parent = Qx))
+      push!(la, map_coefficients(x -> setprecision(Qtok\x, 1), b, parent = Qx))
       push!(lfp, f1*f2)
       i += 2
     end
-    return new(f, map(x -> map_coefficients(y -> setprecision(lift(y, Q), 1), x, parent = Qx), lfp), la, uniformizer(Q), n)
+    return new(f, map(x -> map_coefficients(y -> setprecision(Qtok\y, 1), x, parent = Qx), lfp), la, uniformizer(Q), n)
   end
 
   function HenselCtxdr{S}(f::PolyRingElem{S}) where S

src/LocalField/neq.jl

@@ -27,8 +27,8 @@
 ########### any_root computes a single root in the finite field extensions####
 
 import Nemo: any_root
-function any_root(F::Union{fqPolyRepField, Hecke.RelFinField}, f::Union{fpPolyRingElem, fqPolyRepPolyRingElem})
-   g = polynomial(F, [coeff(f,i) for i = 0:degree(f) ] )
+function any_root(F::Union{fqPolyRepField, Hecke.RelFinField, FqField}, f::PolyElem)
+   g = change_base_ring(F, f; cached = false)
    return any_root(g)
 end
 
@@ -83,11 +83,12 @@
 
 function Nemo.basis(K::FinField, k::FinField)
   b = basis(K)
-  K = base_ring(K)
+  K = base_field(K)
   while absolute_degree(K) > absolute_degree(k)
     b = [x*y for x = basis(K) for y = b]
-    K = base_ring(K)
+    K = base_field(K)
   end
+  @show K, k
   if K != k
     error("subfield not in tower")
   end
@@ -139,7 +140,7 @@
 
   k, mk = residue_field(K)
   @assert absolute_degree(k) == f
-  omega = basis(k, prime_field(k))
+  omega = absolute_basis(k)
   @assert isone(omega[1]) #this has to change...
   mu_0 = valuation(e, p)+1
   e_0 = divexact(e, (p-1)*p^(mu_0-1))
@@ -192,7 +193,7 @@
   k, mk = residue_field(K)
   @assert absolute_degree(k) == f
 
-  b = [preimage(mk, x) for x = basis(k, prime_field(k))]
+  b = [preimage(mk, x) for x = absolute_basis(k)]
   F_K = [ lambda for lambda = 1:ceil(Int, p*e//(p-1))-1 if lambda % p != 0]
   @assert length(F_K) == e
 
@@ -412,15 +413,16 @@
 Find an element `x` in `F` such that the norm from `F` down to the parent of
 `b` is exactly `b`.
 """
-function norm_equation(F::Union{fqPolyRepField, Hecke.RelFinField}, b::Union{fpFieldElem, fqPolyRepFieldElem})
+function norm_equation(F::Union{fqPolyRepField, Hecke.RelFinField, FqField}, b::Union{fpFieldElem, fqPolyRepFieldElem, FqFieldElem})
    if iszero(b)
       return zero(F)
    end
    k = parent(b)
-   n = degree(F,k)
-   f = polynomial(k,vcat([b],[rand(k) for i = 1:n-1],[1]))
+   n = degree(F, k)
+   kt, = polynomial_ring(k, "t", cached = false)
+   f = kt(vcat([b],[rand(k) for i = 1:n-1],[one(k)]))
    while !is_irreducible(f)
-      f = polynomial(k,vcat([b],[rand(k) for i = 1:n-1],[1]))
+     f = kt(vcat([b],[rand(k) for i = 1:n-1],[one(k)]))
    end
    return (-1)^(n)*any_root(F, f)
 end
@@ -796,6 +798,7 @@
   rE, mE = residue_field(E)
   rL, mL = residue_field(L)
   rK, mK = residue_field(K)
+  # how is this supposed to work?
   mrKL = hom(rK, rL, mL(mKL(preimage(mK, gen(rK)))))
   q = order(rK)
 

src/LocalField/qAdic.jl

@@ -10,25 +10,25 @@ function residue_field(Q::FlintQadicField)
   if z !== nothing
     return codomain(z), z
   end
-  Fp = Native.GF(Int(prime(Q)))
+  Fp = Nemo._GF(prime(Q))
   Fpt = polynomial_ring(Fp, cached = false)[1]
   g = defining_polynomial(Q) #no Conway if parameters are too large!
   f = Fpt([Fp(lift(coeff(g, i))) for i=0:degree(Q)])
-  k = Native.finite_field(f, "o", cached = false)[1]
+  k, = Nemo._residue_field(f, "o")
   pro = function(x::qadic)
     v = valuation(x)
     v < 0 && error("elt non integral")
     v > 0 && return k(0)
-    z = k()
+    _z = Fpt()
     for i=0:degree(Q)
-      setcoeff!(z, i, UInt(lift(coeff(x, i))%prime(Q)))
+      setcoeff!(_z, i, Fp(lift(coeff(x, i))))
     end
-    return z
+    return k(_z)
   end
-  lif = function(x::fqPolyRepFieldElem)
+  lif = function(x::FqFieldElem)
     z = Q()
     for i=0:degree(Q)-1
-      setcoeff!(z, i, coeff(x, i))
+      setcoeff!(z, i, lift(ZZ, coeff(x, i)))
     end
     return z
   end
@@ -38,16 +38,16 @@ function residue_field(Q::FlintQadicField)
 end
 
 function residue_field(Q::FlintPadicField)
-  k = Native.GF(Int(prime(Q)))
+  k = Nemo._GF(prime(Q))
   pro = function(x::padic)
     v = valuation(x)
     v < 0 && error("elt non integral")
     v > 0 && return k(0)
     z = k(lift(x))
     return z
   end
-  lif = function(x::fpFieldElem)
-    z = Q(lift(x))
+  lif = function(x::FqFieldElem)
+    z = Q(lift(ZZ, x))
     return z
   end
   return k, MapFromFunc(Q, k, pro, lif)

src/Map/FiniteField.jl

@@ -1,13 +1,19 @@
 function hom(F::FinField, K::FinField, a::FinFieldElem; check::Bool = true)
   @assert parent(a) == K
 
+  # I will be jumping through a lot of hoops to make
+  # base_field(F) == F_p work
   if check
-    @assert iszero(defining_polynomial(F)(a))
+    if absolute_degree(base_field(F)) == 1 || base_field(F) !== base_field(K)
+      @assert iszero(map_coefficients(x -> base_field(K)(lift(ZZ, x)), defining_polynomial(F), cached = false)(a))
+    else
+      @assert iszero(defining_polynomial(F)(a))
+    end
   end
 
   if F isa FqField
     @assert K isa FqField
-    @assert base_field(F) === base_field(K)
+    @assert absolute_degree(F) == 1 || base_field(F) === base_field(K)
     k = base_field(F)
     kx = parent(defining_polynomial(F))
 

src/Misc/FiniteField.jl

@@ -214,6 +214,44 @@
   return phi
 end
 
+function find_morphism(k::FqField, K::FqField)
+  if degree(k) > 1
+    phi = Nemo.find_morphism(k, K) #avoids embed - which stores the info
+  else
+    phi = MapFromFunc(k, K, x -> K(lift(ZZ, x)), y -> k(lift(ZZ, y)))
+  end
+  return phi
+end
+
+function find_morphism(k::FqField, K::fqPolyRepField)
+  # This is no fun
+  if absolute_degree(k) == 1
+    #@assert degree(K) == 1
+    pre = function(x)
+      @assert all(is_zero(coeff(x, i)) for i in 1:(degree(K) - 1))
+      return k(coeff(x, 0))
+    end
+    return MapFromFunc(k, K, x -> K(lift(ZZ, x)), pre)
+  end
+
+  # build K as FqField, then find isomorphism, then go back
+
+  f = modulus(K)
+  a = gen(K)
+  F = prime_field(k)
+  Ft, t = polynomial_ring(F, "t", cached = false)
+  fF = map_coefficients(x -> F(lift(x)), f, parent = Ft)
+  KK, polytoKK = Nemo._residue_field(fF)
+
+  KtoKK = x -> polytoKK(map_coefficients(x -> F(lift(x)), parent(f)(x), parent = Ft))
+
+  KKtoK = x -> K(map_coefficients(x -> coefficient_ring(parent(f))(lift(ZZ, x)), polytoKK\x, parent = parent(f)))
+
+  phi_k_to_KK = Nemo.embed_any(k, KK)
+
+  phi = MapFromFunc(k, K, x -> KKtoK(phi_k_to_KK(x)), x -> phi_k_to_KK\(KtoKK(x)))
+end
+
 
 mutable struct VeryBad
   entries::Ptr{Nothing}

src/NumField/NfAbs/PolyFact.jl

@@ -8,7 +8,7 @@ mutable struct HenselCtxQadic <: Hensel
   n::Int
   #TODO: lift over subfields first iff poly is defined over subfield
   #TODO: use flint if qadic = padic!!
-  function HenselCtxQadic(f::PolyRingElem{qadic}, lfp::Vector{fqPolyRepPolyRingElem})
+  function HenselCtxQadic(f::PolyRingElem{qadic}, lfp::Vector{FqPolyRingElem})
     @assert sum(map(degree, lfp)) == degree(f)
     Q = base_ring(f)
     Qx = parent(f)

src/NumFieldOrd/NfOrd/Clgp/Proof.jl

@@ -84,7 +84,7 @@ function _prime_partition(do_it, nt)
   end
   @assert first(do_it) == 1
   ub = last(do_it)
-  np = ceil(Int, logarithmic_integral(1.0*ub))
+  np = ceil(Int, max(10.0, logarithmic_integral(1.0*ub)))
   primes_per_thread = ceil(Int, np//nt)
   intervals = collect(Iterators.partition(1:np, primes_per_thread))
   res = UnitRange{Int}[]