\import Algebra.Domain
\import Algebra.Field
\import Algebra.Field.Splitting
\import Algebra.Group
\import Algebra.Group.Category
\import Algebra.LatticeColimit
\import Algebra.Monoid
\import Algebra.Monoid.Category
\import Algebra.Pointed
\import Algebra.Pointed.Category
\import Algebra.Ring
\import Algebra.Ring.Category
\import Algebra.Ring.RingHom
\import Algebra.Ring.Integral
\import Algebra.Ring.Poly
\import Algebra.Semiring
\import Arith.Nat
\import Category
\import Data.Array
\import Function.Meta
\import Logic
\import Logic.Meta
\import Meta
\import Paths
\import Paths.Meta
\import Relation.Equivalence
\import Set
\import Set.Category
\import Set.Countable

-- | A ring {K} is algebraically closed if every monic polynomial over {K} of non-zero degree has a root.
\type IsAlgebraicallyClosed (K : Ring)
=> \Pi {p : Poly K} ->  (n : Nat) (n /= 0) (degree<= p n) (polyCoef p n = 1) ->  (a : K) (polyEval p a = 0)

{- | The following conditions are equivalent for a ring {K}:
-   1. {K} is algebraically closed.
-   2. Every monic polynomial over {K} splits into linear factors.
-   2. Every monic polynomial of degree n over {K} splits into n linear factors.
-}
\lemma algebraicallyClosed<->split {K : CRing} : TFAE (
IsAlgebraicallyClosed K,
\Pi {p : Poly K} -> isMonic p ->  (l : Array K) (p = Monoid.BigProd (map (\lam a => padd 1 (negative a)) l)),
\Pi {p : Poly K} (n : Nat) -> degree<= p n -> polyCoef p n = 1 ->  (l : Array K n) (p = Monoid.BigProd (map (\lam a => padd 1 (negative a)) l))
) => TFAE.cycle (
\lam ac (inP (n,pd,pc)) => aux ac pd pc,
\lam c n pd pc => \case c (inP (n,pd,pc)) \with {
| inP (l : Array, q) => \case decideEq n l.len \with {
| yes e => rewrite e $inP (\new l, q) | no r => \have t x => inv$ inv K.zro_*-right *> pmap (x *) (monic-unique r pd pc
(rewrite q $degree<=_BigProd1 \lam _ => idp) (rewrite q$ leadCoef_BigProd1 (\lam _ => idp) \lam _ => idp)) *> ide-right
\in inP (replicate n K.zro, Poly.poly-trivial t _ *> inv (Poly.poly-trivial t _))
}
},
\lam c => \case __ \with {
| inP (0, n/=0, pd, pc) => absurd (n/=0 idp)
| inP (suc n, _, pd, pc) => TruncP.map (c (suc n) pd pc) \lam s => (s.1 0, rewrite s.2 $func-* {polyEvalRingHom _} {padd 1 (negative (s.1 0))} *> pmap (* _) (simplify simplify) *> K.zro_*-left) }) \where { \private \lemma aux {K : CRing} (ac : IsAlgebraicallyClosed K) {p : Poly K} {n : Nat} (pd : degree<= p n) (pc : polyCoef p n = 1) : (l : Array K n) (p = Monoid.BigProd (map (\lam a => padd 1 (negative a)) l)) \elim n | 0 => inP (nil, degree<=0 p pd *> pmap (padd pzero) pc) | suc n => \case ac (inP (suc n, \case __ \with {}, pd, pc)) \with { | inP (a,pa) => \have pd' : degree<= (rootDiv p a) n => degree-monic-reduce _ _ n$ rewrite (rootDiv_* pa) in pd
\in \case aux ac pd' (inv ide-right *> inv (leadCoef-product {_} {_} {_} {n} {1} pd' idp) *> (rewrite (rootDiv_* pa) in pc)) \with {
| inP r => inP (a :: r.1, rootDiv_* pa *> pmap (* _) r.2 *> *-comm)
}
}
}

{- | An algebraically closed ring is integrally closed in any extension which is a strict domain.
-
-   In particular, this implies that such a ring does not have non-trivial integral extensions which are strict domains.
-   Also, this implies that algebraically closed strict domains are integrally closed (in their field of fractions).
-}
\lemma algebraicallyClosed-integrallyClosed {K : CRing} (ac : IsAlgebraicallyClosed K) {E : StrictIntegralDomain} (f : RingHom K E) : isIntegrallyClosed f
=> \lam a => \case __ \with {
| inP (p, inP (n, pd, pc), pe) => \case algebraicallyClosed<->split 0 2 ac n pd pc \with {
| inP s => \case E.zeroBigProd $inv (MonoidHom.func-BigProd {polyMapEvalRingHom f a}) *> (rewrite s.2 in pe) \with { | inP (j,q) => inP (s.1 j, inv$ E.fromZero $simplify (rewrite (f.func-ide,f.func-negative) simplify) *> q) } } } \type IsAlgebraicClosure (f : RingHom) => \Sigma (IsAlgebraicallyClosed f.Cod) (isIntegralExt f) -- | If E is a strict domain, f : K -> E is integral, and every monic polynomial in {K} splits into linear factors in {E}, then {f} is an algebraic closure. \lemma algebraicClosure-split {K : CRing} {E : StrictIntegralDomain} (f : RingHom K E) (fi : isIntegralExt f) (c : \Pi {p : Poly K} -> isMonic p -> (l : Array E) (polyMap f p = Monoid.BigProd (map (\lam a => padd 1 (negative a)) l))) : IsAlgebraicClosure f => (\lam {p} (inP (n,n/=0,pd,pc)) => \case integralExt_polyDiv fi (inP (n,pd,pc)) \with { | inP (q, qm, p|q : Monoid.LDiv) => \case c qm \with { | inP (l,r) => \case polyDiv-linear (*-comm *> p|q.inv-right *> r) \with { | byLeft r => inP (l r.1, r.2) | byRight r => absurd \case polyInv r, \elim n, \elim n/=0, \elim pc \with { | _, 0, n/=0, pc => n/=0 idp | inP t, suc n, _, pc => E.zro/=ide$ inv (pmap (polyCoef __ _) t.2) *> pc
}
}
}
}, fi)

-- | Every countable discrete field has an algebraic closure.
\sfunc countableAlgebraicClosure {k : DiscreteField} (c : Countable k) : \Sigma (K : DiscreteField) (f : RingHom k K) (IsAlgebraicClosure f)
=> \let | ps n => pointed-countable zro (Poly.poly-countable c) n
| S => sequence c ps
| sf n => countableSplittingField (S n).2 {polyMap (S n).3 (ps n)}
| F => NatFunctor (\lam n => (S n).1) \lam {n} => (sf n).3
| E => DiscreteFieldLatticeColimit {NatBSemilattice} F \lam n => (S n).1
| h n => RingLatticeColimit.inMap {_} {F} n
\in (E, h 0, algebraicClosure-split {_} {E} _ (later \lam (in~ (n,a)) =>
\let t => integral_image-comp (RingLatticeColimit.inMap {_} {F} n) ((S n).4 a)
\in transportInv {RingHom k E} (isIntegral __ _) (later $exts \lam a => SetColimit.~-cequiv zero<=_$ pmap (NatFunctor.natHom {CRingCat} __ _) (NatFunctor.sigma-isProp _ _) *> sequence3) t)
\lam {p} pm => \case pointed-countable-surj zro (Poly.poly-countable c) p \with {
| inP (n,q) => \case splitting-monic {_} {_} {polyMap (S n).3 (ps n)} (rewrite q $polyMap_isMonic pm) (sf n).4 \with { | inP (l,_,lq) => inP (map (h (suc n)) l, later rewrite q ( pmap {RingHom (F 0) (CRingLatticeColimit F)} (polyMap __ p) {h 0} {h (suc n) ((sf n).3 (S n).3)} (exts \lam a => SetColimit.~-cequiv zero<=_$ pmap (sf n).3 $pmap (NatFunctor.natHom {CRingCat} __ a) (NatFunctor.sigma-isProp _ _) *> sequence3) *> inv (polyMap-comp _ _) *> pmap (polyMap _) (inv (polyMap-comp _ _))) *> pmap (polyMapRingHom _) lq *> MonoidHom.func-BigProd {polyMapRingHom (h (suc n))} *> pmap Monoid.BigProd (exts \lam j => pmap2 (\lam x y => padd (padd pzero x) y) func-ide AddGroupHom.func-negative)) } }) \where \private { \func sequence {k : DiscreteField} (kc : Countable k) (f : Nat -> Poly k) (n : Nat) : \Sigma (K : DiscreteField) (Countable K) (g : RingHom k K) (isIntegralExt g) \elim n | 0 => (k, kc, id k, integralExt_id) | suc n => \let | (K,Kc,g,gi) => sequence kc f n | t => countableSplittingField Kc {polyMap g (f n)} \in (t.1, t.2, t.3 RingCat. g, integralExt-comp _ _ gi$ splitting-integral (countableSplittingField Kc).4)

\lemma sequence3 {k : DiscreteField} {kc : Countable k} {f : Nat -> Poly k} {n : Nat} {a : k}
: NatFunctor.natHom {CRingCat} {_} {\lam {n} => (countableSplittingField (sequence kc f n).2 {polyMap (sequence kc f n).3 (f n)}).3} {0} (n,idp) a = (sequence kc f n).3 a \elim n
| 0 => idp
| suc n => pmap (countableSplittingField (sequence kc f n).2).3 sequence3
}