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ring_theory.polynomial.cyclotomic

Cyclotomic polynomials. #

For n : ℕ and an integral domain R, we define a modified version of the n-th cyclotomic polynomial with coefficients in R, denoted cyclotomic' n R, as ∏ (X - μ), where μ varies over the primitive nth roots of unity. If there is a primitive nth root of unity in R then this the standard definition. We then define the standard cyclotomic polynomial cyclotomic n R with coefficients in any ring R.

Main definition #

Main results #

Implementation details #

Our definition of cyclotomic' n R makes sense in any integral domain R, but the interesting results hold if there is a primitive n-th root of unity in R. In particular, our definition is not the standard one unless there is a primitive nth root of unity in R. For example, cyclotomic' 3 ℤ = 1, since there are no primitive cube roots of unity in . The main example is R = ℂ, we decided to work in general since the difficulties are essentially the same. To get the standard cyclotomic polynomials, we use int_coeff_of_cycl, with R = ℂ, to get a polynomial with integer coefficients and then we map it to polynomial R, for any ring R. To prove cyclotomic.irreducible, the irreducibility of cyclotomic n ℤ, we show in cyclotomic_eq_minpoly that cyclotomic n ℤ is the minimal polynomial of any n-th primitive root of unity μ : K, where K is a field of characteristic 0.

noncomputable def polynomial.cyclotomic' (n : ) (R : Type u_1) [comm_ring R] [is_domain R] :

The modified n-th cyclotomic polynomial with coefficients in R, it is the usual cyclotomic polynomial if there is a primitive n-th root of unity in R.

Equations
@[simp]
theorem polynomial.cyclotomic'_zero (R : Type u_1) [comm_ring R] [is_domain R] :

The zeroth modified cyclotomic polyomial is 1.

@[simp]

The first modified cyclotomic polyomial is X - 1.

@[simp]
theorem polynomial.cyclotomic'_two (R : Type u_1) [comm_ring R] [is_domain R] (p : ) [char_p R p] (hp : p 2) :

The second modified cyclotomic polyomial is X + 1 if the characteristic of R is not 2.

theorem polynomial.cyclotomic'.monic (n : ) (R : Type u_1) [comm_ring R] [is_domain R] :

cyclotomic' n R is monic.

theorem polynomial.cyclotomic'_ne_zero (n : ) (R : Type u_1) [comm_ring R] [is_domain R] :

cyclotomic' n R is different from 0.

theorem polynomial.nat_degree_cyclotomic' {R : Type u_1} [comm_ring R] [is_domain R] {ζ : R} {n : } (h : is_primitive_root ζ n) :

The natural degree of cyclotomic' n R is totient n if there is a primitive root of unity in R.

theorem polynomial.degree_cyclotomic' {R : Type u_1} [comm_ring R] [is_domain R] {ζ : R} {n : } (h : is_primitive_root ζ n) :

The degree of cyclotomic' n R is totient n if there is a primitive root of unity in R.

The roots of cyclotomic' n R are the primitive n-th roots of unity.

theorem polynomial.X_pow_sub_one_eq_prod {K : Type u_1} [field K] {ζ : K} {n : } (hpos : 0 < n) (h : is_primitive_root ζ n) :

If there is a primitive nth root of unity in K, then X ^ n - 1 = ∏ (X - μ), where μ varies over the n-th roots of unity.

cyclotomic' n K splits.

theorem polynomial.X_pow_sub_one_splits {K : Type u_1} [field K] {ζ : K} {n : } (h : is_primitive_root ζ n) :

If there is a primitive n-th root of unity in K, then X ^ n - 1splits.

theorem polynomial.prod_cyclotomic'_eq_X_pow_sub_one {K : Type u_1} [field K] {ζ : K} {n : } (hpos : 0 < n) (h : is_primitive_root ζ n) :

If there is a primitive n-th root of unity in K, then ∏ i in nat.divisors n, cyclotomic' i K = X ^ n - 1.

theorem polynomial.cyclotomic'_eq_X_pow_sub_one_div {K : Type u_1} [field K] {ζ : K} {n : } (hpos : 0 < n) (h : is_primitive_root ζ n) :

If there is a primitive n-th root of unity in K, then cyclotomic' n K = (X ^ k - 1) /ₘ (∏ i in nat.proper_divisors k, cyclotomic' i K).

If there is a primitive n-th root of unity in K, then cyclotomic' n K comes from a monic polynomial with integer coefficients.

If K is of characteristic 0 and there is a primitive n-th root of unity in K, then cyclotomic n K comes from a unique polynomial with integer coefficients.

noncomputable def polynomial.cyclotomic (n : ) (R : Type u_1) [ring R] :

The n-th cyclotomic polynomial with coefficients in R.

Equations

cyclotomic n R comes from cyclotomic n ℤ.

@[simp]
theorem polynomial.map_cyclotomic (n : ) {R : Type u_1} {S : Type u_2} [ring R] [ring S] (f : R →+* S) :

The definition of cyclotomic n R commutes with any ring homomorphism.

@[simp]
theorem polynomial.cyclotomic_zero (R : Type u_1) [ring R] :

The zeroth cyclotomic polyomial is 1.

@[simp]
theorem polynomial.cyclotomic_one (R : Type u_1) [ring R] :

The first cyclotomic polyomial is X - 1.

@[simp]
theorem polynomial.cyclotomic_two (R : Type u_1) [ring R] :

The second cyclotomic polyomial is X + 1.

theorem polynomial.cyclotomic.monic (n : ) (R : Type u_1) [ring R] :

cyclotomic n is monic.

theorem polynomial.cyclotomic_ne_zero (n : ) (R : Type u_1) [ring R] [nontrivial R] :

cyclotomic n R is different from 0.

theorem polynomial.degree_cyclotomic (n : ) (R : Type u_1) [ring R] [nontrivial R] :

The degree of cyclotomic n is totient n.

The natural degree of cyclotomic n is totient n.

theorem polynomial.degree_cyclotomic_pos (n : ) (R : Type u_1) (hpos : 0 < n) [ring R] [nontrivial R] :

The degree of cyclotomic n R is positive.

theorem polynomial.prod_cyclotomic_eq_X_pow_sub_one {n : } (hpos : 0 < n) (R : Type u_1) [comm_ring R] :

∏ i in nat.divisors n, cyclotomic i R = X ^ n - 1.

theorem is_root_of_unity_iff {n : } (h : 0 < n) (R : Type u_1) [comm_ring R] [is_domain R] {ζ : R} :
ζ ^ n = 1 ∃ (i : ) (H : i n.divisors), (polynomial.cyclotomic i R).is_root ζ

cyclotomic n R can be expressed as a product in a fraction field of polynomial R using Möbius inversion.

theorem polynomial.cyclotomic_eq_X_pow_sub_one_div {R : Type u_1} [comm_ring R] {n : } (hpos : 0 < n) :

We have cyclotomic n R = (X ^ k - 1) /ₘ (∏ i in nat.proper_divisors k, cyclotomic i K).

theorem polynomial.X_pow_sub_one_dvd_prod_cyclotomic (R : Type u_1) [comm_ring R] {n m : } (hpos : 0 < n) (hm : m n) (hdiff : m n) :

If m is a proper divisor of n, then X ^ m - 1 divides ∏ i in nat.proper_divisors n, cyclotomic i R.

theorem polynomial.cyclotomic_eq_prod_X_sub_primitive_roots {K : Type u_1} [field K] {ζ : K} {n : } (hz : is_primitive_root ζ n) :

If there is a primitive n-th root of unity in K, then cyclotomic n K = ∏ μ in primitive_roots n R, (X - C μ). In particular, cyclotomic n K = cyclotomic' n K

theorem polynomial.is_root_cyclotomic {n : } {K : Type u_1} [field K] (hpos : 0 < n) {μ : K} (h : is_primitive_root μ n) :

Any n-th primitive root of unity is a root of cyclotomic n K.

theorem polynomial.is_root_cyclotomic_iff {n : } {R : Type u_1} [comm_ring R] [is_domain R] {μ : R} (hn : n 0) :
theorem polynomial.eq_cyclotomic_iff {R : Type u_1} [comm_ring R] {n : } (hpos : 0 < n) (P : polynomial R) :

If p is prime, then cyclotomic p R = geom_sum X p.

theorem polynomial.cyclotomic_prime_pow_eq_geom_sum {R : Type u_1} [comm_ring R] {p n : } (hp : nat.prime p) :

If p ^ k is prime power, then cyclotomic (p ^ (n + 1)) R = geom_sum (X ^ p ^ n) p.

theorem polynomial.cyclotomic_coeff_zero (R : Type u_1) [comm_ring R] {n : } (hn : 2 n) :

The constant term of cyclotomic n R is 1 if 2 ≤ n.

theorem polynomial.coprime_of_root_cyclotomic {n : } (hpos : 0 < n) {p : } [hprime : fact (nat.prime p)] {a : } (hroot : (polynomial.cyclotomic n (zmod p)).is_root ((nat.cast_ring_hom (zmod p)) a)) :

If (a : ℕ) is a root of cyclotomic n (zmod p), where p is a prime, then a and p are coprime.

theorem polynomial.order_of_root_cyclotomic_dvd {n : } (hpos : 0 < n) {p : } [fact (nat.prime p)] {a : } (hroot : (polynomial.cyclotomic n (zmod p)).is_root ((nat.cast_ring_hom (zmod p)) a)) :

If (a : ℕ) is a root of cyclotomic n (zmod p), then the multiplicative order of a modulo p divides n.

theorem minpoly_dvd_cyclotomic {n : } {K : Type u_1} [field K] {μ : K} (h : is_primitive_root μ n) (hpos : 0 < n) [char_zero K] :

The minimal polynomial of a primitive n-th root of unity μ divides cyclotomic n ℤ.

theorem polynomial.cyclotomic_eq_minpoly {n : } {K : Type u_1} [field K] {μ : K} (h : is_primitive_root μ n) (hpos : 0 < n) [char_zero K] :

cyclotomic n ℤ is the minimal polynomial of a primitive n-th root of unity μ.

cyclotomic n ℤ is irreducible.

@[simp]
@[simp]
theorem polynomial.eval₂_one_cyclotomic_prime {R : Type u_1} {S : Type u_2} [comm_ring R] [semiring S] (f : R →+* S) {n : } [fact (nat.prime n)] :
@[simp]
theorem polynomial.eval_one_cyclotomic_prime_pow {R : Type u_1} [comm_ring R] {n : } (k : ) [hn : fact (nat.prime n)] :
@[simp]
theorem polynomial.eval₂_one_cyclotomic_prime_pow {R : Type u_1} {S : Type u_2} [comm_ring R] [semiring S] (f : R →+* S) {n : } (k : ) [fact (nat.prime n)] :