SchurTruncatedExponential.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 1 contain sorry, 0 declared as axioms

Schur's theorem on Galois groups of truncated exponential polynomials

Reference: (https://math.stackexchange.com/questions/2814220)

Reference (https://mathoverflow.net/questions/477077)

Imports (1)

FormalConjecturesUtil

def SchurTruncatedExponential.truncatedExp Documented declaration: truncatedExp line 45 source
Source documentation
The truncated exponential polynomial truncatedExp n is given by ∑_{j=0}^{n} x^j / j! over ℚ, which is the n-th partial sum of the Taylor series for the exponential function e^x.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Polynomial, Rat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, OfNat.ofNat, HSMul.hSMul, HDiv.hDiv, Nat.factorial, HPow.hPow, Polynomial.X
21 instances, coercions and auxiliary constants left out.
```
noncomputable def truncatedExp (n : ℕ) : ℚ[X] :=
…
```
theorem SchurTruncatedExponential.schur_truncatedExp_galoisGroup_equiv Schur's Theorem (1924): Let f_n(x) = ∑_{j=0}^n x^j/j! be the n-th truncated exponential polynomial over ℚ sorry line 57 source
Source documentation
Schur's Theorem (1924): Let f_n(x) = ∑_{j=0}^n x^j/j! be the n-th truncated exponential polynomial over ℚ. Then for n ≥ 2:
- If n ≡ 0 (mod 4), the Galois group of f_n is isomorphic to the alternating group A_n
- If n ≢ 0 (mod 4), the Galois group of f_n is isomorphic to the symmetric group S_n
depends on sorry what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, ite, Eq, HMod.hMod, Nonempty, MulEquiv, Polynomial.Gal, Rat, SchurTruncatedExponential.truncatedExp, Subtype, Equiv.Perm, Fin, Membership.mem, Subgroup, alternatingGroup
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem schur_truncatedExp_galoisGroup_equiv (n : ℕ) (hn : n ≥ 2) :
  letI f := truncatedExp n
…
```
Mentions: SchurTruncatedExponential.truncatedExp