# mathlib3documentation

ring_theory.polynomial.eisenstein.is_integral

# Eisenstein polynomials #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4. In this file we gather more miscellaneous results about Eisenstein polynomials

## Main results #

• mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at: let K be the field of fraction of an integrally closed domain R and let L be a separable extension of K, generated by an integral power basis B such that the minimal polynomial of B.gen is Eisenstein at p. Given z : L integral over R, if p ^ n • z ∈ adjoin R {B.gen}, then z ∈ adjoin R {B.gen}. Together with algebra.discr_mul_is_integral_mem_adjoin this result often allows to compute the ring of integers of L.
theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_is_eiseinstein_at {R : Type u} {K : Type v} {L : Type z} {p : R} [comm_ring R] [field K] [field L] [ L] [ L] [ K] [ L] [ L] [is_domain R] [ K] {B : L} (hp : prime p) (hBint : B.gen) {z : L} {Q : polynomial R} (hQ : Q = p z) (hzint : z) (hei : (minpoly R B.gen).is_eisenstein_at {p})) :
p Q.coeff 0

Let K be the field of fraction of an integrally closed domain R and let L be a separable extension of K, generated by an integral power basis B such that the minimal polynomial of B.gen is Eisenstein at p. Given z : L integral over R, if Q : R[X] is such that aeval B.gen Q = p • z, then p ∣ Q.coeff 0.

theorem mem_adjoin_of_dvd_coeff_of_dvd_aeval {A : Type u_1} {B : Type u_2} [comm_ring B] [ B] {Q : polynomial A} {p : A} {x z : B} (hp : p 0) (hQ : (i : ), i finset.range (Q.nat_degree + 1) p Q.coeff i) (hz : Q = p z) :
z {x}
theorem mem_adjoin_of_smul_prime_smul_of_minpoly_is_eiseinstein_at {R : Type u} {K : Type v} {L : Type z} {p : R} [comm_ring R] [field K] [field L] [ L] [ L] [ K] [ L] [ L] [is_domain R] [ K] {B : L} (hp : prime p) (hBint : B.gen) {z : L} (hzint : z) (hz : p z {B.gen}) (hei : (minpoly R B.gen).is_eisenstein_at {p})) :
z {B.gen}

Let K be the field of fraction of an integrally closed domain R and let L be a separable extension of K, generated by an integral power basis B such that the minimal polynomial of B.gen is Eisenstein at p. Given z : L integral over R, if p • z ∈ adjoin R {B.gen}, then z ∈ adjoin R {B.gen}.

theorem mem_adjoin_of_smul_prime_pow_smul_of_minpoly_is_eiseinstein_at {R : Type u} {K : Type v} {L : Type z} {p : R} [comm_ring R] [field K] [field L] [ L] [ L] [ K] [ L] [ L] [is_domain R] [ K] {B : L} (hp : prime p) (hBint : B.gen) {n : } {z : L} (hzint : z) (hz : p ^ n z {B.gen}) (hei : (minpoly R B.gen).is_eisenstein_at {p})) :
z {B.gen}

Let K be the field of fraction of an integrally closed domain R and let L be a separable extension of K, generated by an integral power basis B such that the minimal polynomial of B.gen is Eisenstein at p. Given z : L integral over R, if p ^ n • z ∈ adjoin R {B.gen}, then z ∈ adjoin R {B.gen}. Together with algebra.discr_mul_is_integral_mem_adjoin this result often allows to compute the ring of integers of L.