Documentation

Mathlib.Algebra.CharP.Basic

Characteristic of semirings #

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theorem Commute.add_pow_prime_pow_eq {R : Type u_1} [Semiring R] {p : ℕ} {x : R} {y : R} (hp : Nat.Prime p) (h : Commute x y) (n : ℕ) :
(x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n + ↑p * ∑ k ∈ Finset.Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * ↑((p ^ n).choose k / p)
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theorem Commute.add_pow_prime_eq {R : Type u_1} [Semiring R] {p : ℕ} {x : R} {y : R} (hp : Nat.Prime p) (h : Commute x y) :
(x + y) ^ p = x ^ p + y ^ p + ↑p * ∑ k ∈ Finset.Ioo 0 p, x ^ k * y ^ (p - k) * ↑(p.choose k / p)
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theorem Commute.exists_add_pow_prime_pow_eq {R : Type u_1} [Semiring R] {p : ℕ} {x : R} {y : R} (hp : Nat.Prime p) (h : Commute x y) (n : ℕ) :
∃ (r : R), (x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n + ↑p * r
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theorem Commute.exists_add_pow_prime_eq {R : Type u_1} [Semiring R] {p : ℕ} {x : R} {y : R} (hp : Nat.Prime p) (h : Commute x y) :
∃ (r : R), (x + y) ^ p = x ^ p + y ^ p + ↑p * r
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theorem add_pow_prime_pow_eq {R : Type u_1} [CommSemiring R] {p : ℕ} (hp : Nat.Prime p) (x : R) (y : R) (n : ℕ) :
(x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n + ↑p * ∑ k ∈ Finset.Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * ↑((p ^ n).choose k / p)
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theorem add_pow_prime_eq {R : Type u_1} [CommSemiring R] {p : ℕ} (hp : Nat.Prime p) (x : R) (y : R) :
(x + y) ^ p = x ^ p + y ^ p + ↑p * ∑ k ∈ Finset.Ioo 0 p, x ^ k * y ^ (p - k) * ↑(p.choose k / p)
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theorem exists_add_pow_prime_pow_eq {R : Type u_1} [CommSemiring R] {p : ℕ} (hp : Nat.Prime p) (x : R) (y : R) (n : ℕ) :
∃ (r : R), (x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n + ↑p * r
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theorem exists_add_pow_prime_eq {R : Type u_1} [CommSemiring R] {p : ℕ} (hp : Nat.Prime p) (x : R) (y : R) :
∃ (r : R), (x + y) ^ p = x ^ p + y ^ p + ↑p * r
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theorem add_pow_char_of_commute (R : Type u_1) [Semiring R] {p : ℕ} [hp : Fact (Nat.Prime p)] [CharP R p] (x : R) (y : R) (h : Commute x y) :
(x + y) ^ p = x ^ p + y ^ p
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theorem add_pow_char_pow_of_commute (R : Type u_1) [Semiring R] {p : ℕ} {n : ℕ} [hp : Fact (Nat.Prime p)] [CharP R p] (x : R) (y : R) (h : Commute x y) :
(x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n
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theorem sub_pow_char_of_commute (R : Type u_1) [Ring R] {p : ℕ} [Fact (Nat.Prime p)] [CharP R p] (x : R) (y : R) (h : Commute x y) :
(x - y) ^ p = x ^ p - y ^ p
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theorem sub_pow_char_pow_of_commute (R : Type u_1) [Ring R] {p : ℕ} [Fact (Nat.Prime p)] [CharP R p] {n : ℕ} (x : R) (y : R) (h : Commute x y) :
(x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n
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theorem add_pow_char (R : Type u_1) [CommSemiring R] {p : ℕ} [Fact (Nat.Prime p)] [CharP R p] (x : R) (y : R) :
(x + y) ^ p = x ^ p + y ^ p
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theorem add_pow_char_pow (R : Type u_1) [CommSemiring R] {p : ℕ} [Fact (Nat.Prime p)] [CharP R p] {n : ℕ} (x : R) (y : R) :
(x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n
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theorem sub_pow_char (R : Type u_1) [CommRing R] {p : ℕ} [Fact (Nat.Prime p)] [CharP R p] (x : R) (y : R) :
(x - y) ^ p = x ^ p - y ^ p
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theorem sub_pow_char_pow (R : Type u_1) [CommRing R] {p : ℕ} [Fact (Nat.Prime p)] [CharP R p] {n : ℕ} (x : R) (y : R) :
(x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n
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theorem CharP.neg_one_pow_char (R : Type u_1) [Ring R] (p : ℕ) [CharP R p] [Fact (Nat.Prime p)] :
(-1) ^ p = -1
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theorem CharP.neg_one_pow_char_pow (R : Type u_1) [Ring R] (p : ℕ) (n : ℕ) [CharP R p] [Fact (Nat.Prime p)] :
(-1) ^ p ^ n = -1
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theorem CharP.char_ne_zero_of_finite (R : Type u_1) [NonAssocRing R] (p : ℕ) [CharP R p] [Finite R] :
p ≠ 0

The characteristic of a finite ring cannot be zero.

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theorem CharP.ringChar_ne_zero_of_finite (R : Type u_1) [NonAssocRing R] [Finite R] :
ringChar R ≠ 0
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theorem CharP.char_is_prime (R : Type u_1) [Ring R] [NoZeroDivisors R] [Nontrivial R] [Finite R] (p : ℕ) [CharP R p] :
Nat.Prime p