`Compfiles/Usa1976P5.lean`

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3 declarations, 0 contain sorry, 0 declared as axioms

USA Mathematical Olympiad 1976, Problem 5

The polynomials a(x), b(x), c(x), d(x) satisfy a(x⁵) + x·b(x⁵) + x²·c(x⁵) = (1 + x + x² + x³ + x⁴)·d(x). Show that a(x) has the factor (x - 1).

Imports (7)

Imported by 1 file

Compfiles.lean

abbrev Usa1976P5.ω A primitive fifth root of unity in ℂ line 36 source
Source documentation
A primitive fifth root of unity in ℂ.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Complex
Proof uses: Complex.exp, HMul.hMul, OfNat.ofNat, Real.pi, Complex.I, Inv.inv
8 instances, coercions and auxiliary constants left out.
```
noncomputable abbrev ω : ℂ := Complex.exp (2 * (Real.pi : ℂ) * Complex.I * (5 : ℂ)⁻¹)
```
lemma Usa1976P5.isPrimitiveRoot_ω line 38 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: IsPrimitiveRoot, Complex, Usa1976P5.ω, OfNat.ofNat, Nat
Proof uses: Complex.exp, HMul.hMul, Real.pi, Complex.I, HDiv.hDiv, Complex.isPrimitiveRoot_exp_of_coprime, Mathlib.Meta.NormNum.isNat_eq_false, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.false, of_decide_eq_true, Nat.Coprime, id, Eq, Decidable.decide, Bool.true, Eq.mp, Inv.inv, congrFun' and 4 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
problem Usa1976P5.usa1976_p5 line 44 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Complex, Eq, HAdd.hAdd, Polynomial.comp, HPow.hPow, Nat, Polynomial.X, OfNat.ofNat, HMul.hMul, Dvd.dvd, HSub.hSub, RingHom, Polynomial.C
Proof uses: Polynomial.eval, rfl, LT.lt, Usa1976P5.ω, Nat.Coprime, Nat.Coprime.symm, Iff.mpr, Not, Nat.Prime.coprime_iff_not_dvd, Nat.prime_five, Nat.not_dvd_of_pos_of_lt, IsPrimitiveRoot, IsPrimitiveRoot.pow_of_coprime, Usa1976P5.isPrimitiveRoot_ω, IsPrimitiveRoot.pow_eq_one, Eq.mp, Finset.sum, Finset.range, congrFun', congrArg and 97 more
139 instances, coercions and auxiliary constants left out.

Tactic trace: 58 steps
Open to load the checked before-and-after goal states.