Section MoreAlgCaut.
Implicit Type rR : unitRingType.
End NumFieldProj.
(* Integral spans. *)
Definition Cint_span (s : seq algC) : pred algC :=
fun x => dec_Cint_span (in_tuple [seq \row_(i < 1) y | y <- s]) (\row_i x).
Fact Cint_span_key s : pred_key (Cint_span s). Proof. by []. Qed.
Canonical Cint_span_keyed s := KeyedPred (Cint_span_key s).
Canonical Cint_span_opprPred s := OpprPred (Cint_span_zmod_closed s).
Canonical Cint_span_addrPred s := AddrPred (Cint_span_zmod_closed s).
Canonical Cint_span_zmodPred s := ZmodPred (Cint_span_zmod_closed s).
(* Algebraic integers. *)
Definition Aint : {pred algC} := fun x => minCpoly x \is a polyOver Cint.
Fact Aint_key : pred_key Aint. Proof. by []. Qed.
Canonical Aint_keyed := KeyedPred Aint_key.

Lemma root_monic_Aint p x :
root p x -> p \is monic -> p \is a polyOver Cint -> x \in Aint.