fix: valuation for inert primes

- also fix Hensel lifting in the trivial case

src/Misc/Poly.jl

@@ -253,6 +253,14 @@ function hensel_lift(f::ZZPolyRingElem, g::ZZPolyRingElem, h::ZZPolyRingElem, p:
   ## is essentially f and f would be a legal answer. Probably reduced mod p^k
   ## with all non-negative coefficients
   ## for now:
+  if is_one(g)
+    h = mod(h, p^k)
+    return g, h
+  elseif is_one(h)
+    g = mod(g, p^k)
+    return g, h
+  end
+
   @assert !iszero(a) && !iszero(b)
   a = lift(parent(g), a)
   b = lift(parent(g), b)

src/NumFieldOrd/NfOrd/Ideal/Valuation.jl

@@ -22,7 +22,12 @@ function val_func_no_index_small(p::AbsNumFieldOrderIdeal{AbsSimpleNumField, Abs
   gR = gcd!(gR, gR, f)
   g = lift(Zx, gR)
   k = flog(ZZRingElem(typemax(UInt)), P)
-  g = hensel_lift(Zx(K.pol), g, P, k)
+  if degree(p) == degree(K)
+    # inert prime, K.pol is irreducible mod p
+    g = Zx(K.pol)
+  else
+    g = hensel_lift(Zx(K.pol), g, P, k)
+  end
   Sx = polynomial_ring(residue_ring(ZZ, UInt(P)^k, cached=false)[1], cached=false)[1]
   g = Sx(g)
   h = Sx()

test/Misc/Poly.jl

@@ -229,4 +229,13 @@ end
   end
 end
 
+@testset "hensel" begin
+  Zx, x = ZZ["x"]
+  f = x^2 + x + 1
+  h = one(Zx)
+  (gg, hh) = hensel_lift(f, f, h, ZZ(2), 2)
+  @test mod(gg * hh, ZZ(4)) == mod(f, ZZ(4))
+  (gg, hh) = hensel_lift(f, h, f, ZZ(2), 2)
+  @test mod(gg * hh, ZZ(4)) == mod(f, ZZ(4))
+end
 

test/NfOrd/Ideal/Prime.jl

@@ -143,3 +143,9 @@ E = pmaximal_overorder(O, 23)
 lp = prime_decomposition(E, 23)
 @test length(lp) == 2
 
+let
+  # valuation for large degree, inert prime
+  K, a = cyclotomic_real_subfield(101, :a)
+  P, = prime_ideals_over(maximal_order(K), 10007)
+  @test valuation(gen(K), P) == 0
+end