# mathlibdocumentation

ring_theory.henselian

# Henselian rings #

In this file we set up the basic theory of Henselian (local) rings. A ring R is Henselian at an ideal I if the following conditions hold:

• I is contained in the Jacobson radical of R
• for every polynomial f over R, with a simple root a₀ over the quotient ring R/I, there exists a lift a : R of a₀ that is a root of f.

(Here, saying that a root b of a polynomial g is simple means that g.derivative.eval b is a unit. Warning: if R/I is not a field then it is not enough to assume that g has a factorization into monic linear factors in which X - b shows up only once; for example 1 is not a simple root of X^2-1 over ℤ/4ℤ.)

A local ring R is Henselian if it is Henselian at its maximal ideal. In this case the first condition is automatic, and in the second condition we may ask for f.derivative.eval a ≠ 0, since the quotient ring R/I is a field in this case.

## Main declarations #

• henselian_ring: a typeclass on commutative rings, asserting that the ring is Henselian at the ideal I.
• henselian_local_ring: a typeclass on commutative rings, asserting that the ring is local Henselian.
• field.henselian: fields are Henselian local rings
• henselian.tfae: equivalent ways of expressing the Henselian property for local rings
• is_adic_complete.henselian: a ring R with ideal I that is I-adically complete is Henselian at I

## References #

https://stacks.math.columbia.edu/tag/04GE

## Todo #

After a good API for etale ring homomorphisms has been developed, we can give more equivalent characterization os Henselian rings.

In particular, this can give a proof that factorizations into coprime polynomials can be lifted from the residue field to the Henselian ring.

The following gist contains some code sketches in that direction. https://gist.github.com/jcommelin/47d94e4af092641017a97f7f02bf9598

theorem is_local_ring_hom_of_le_jacobson_bot {R : Type u_1} [comm_ring R] (I : ideal R) (h : I .jacobson) :
@[class]
structure henselian_ring (R : Type u_1) [comm_ring R] (I : ideal R) :
Prop

A ring R is Henselian at an ideal I if the following condition holds: for every polynomial f over R, with a simple root a₀ over the quotient ring R/I, there exists a lift a : R of a₀ that is a root of f.

(Here, saying that a root b of a polynomial g is simple means that g.derivative.eval b is a unit. Warning: if R/I is not a field then it is not enough to assume that g has a factorization into monic linear factors in which X - b shows up only once; for example 1 is not a simple root of X^2-1 over ℤ/4ℤ.)

Instances of this typeclass
@[class]
structure henselian_local_ring (R : Type u_1) [comm_ring R] :
Prop
• to_local_ring :
• is_henselian : ∀ (f : , f.monic∀ (a₀ : R), (∃ (a : R), f.is_root a a - a₀

A local ring R is Henselian if the following condition holds: for every polynomial f over R, with a simple root a₀ over the residue field, there exists a lift a : R of a₀ that is a root of f. (Recall that a root b of a polynomial g is simple if it is not a double root, so if g.derivative.eval b ≠ 0.)

In other words, R is local Henselian if it is Henselian at the ideal I, in the sense of henselian_ring.

Instances of this typeclass
@[protected, instance]
def field.henselian (K : Type u_1) [field K] :
theorem henselian_local_ring.tfae (R : Type u) [comm_ring R] [local_ring R] :
, ∀ (f : , f.monic∀ (a₀ : , (polynomial.aeval a₀) f = 0 0(∃ (a : R), f.is_root a a = a₀), ∀ {K : Type u} [_inst_3 : field K] (φ : R →+* K), ∀ (f : , f.monic∀ (a₀ : K), a₀ f = 0 0(∃ (a : R), f.is_root a φ a = a₀)].tfae
@[protected, instance]
@[protected, instance]
def is_adic_complete.henselian_ring (R : Type u_1) [comm_ring R] (I : ideal R) [ R] :

A ring R that is I-adically complete is Henselian at I.