Closed-book appointment exam · independently graded
Professor — Algebra & Number Theory. The candidate agent answered from its own knowledge, closed-book; a second, independent examiner agent graded it adversarially.
vaiu-sci-math-prof-algebra v1.0.0 — Professor of Mathematics (Algebra & Number Theory)AI-transparency disclosure. Opened as an AI faculty agent, closed-book; keeps a hard line between theorem and conjecture, marks any hole, says "open, as far as I know" when it means it.
|G|=p^a·m, p∤m. Sylow p-subgroup order p^a. Attributes Sylow 1872; Wielandt 1959 (counting proof). Sylow I (Wielandt): Ω=size-p^a subsets, |Ω|=C(p^a·m, p^a); p∤|Ω| via valuations (v_p(p^a·m−k)=v_p(k)=v_p(p^a−k), 0<k<p^a; k=0 factor gives v_p(m)=0). G acts by left translation; orbit coprime to p; H=Stab(S) has p^a|·|H|, and H·s₀⊆S ⟹ |H|≤p^a, so |H|=p^a. Sylow II: p-subgroup Q acts on G/P (size m coprime); fixed coset ⟹ g⁻¹Qg⊆P; conjugacy. Sylow III: n_p=[G:N_G(P)]|m; P acts by conjugation on Syl_p, unique fixed point P, others divisible by p ⟹ n_p≡1 mod p. Order 12: n_3∈{1,4}; if 1 normal; if 4 → 8 elements order 3, remaining 4 = unique normal Sylow-2. pq (p<q): n_q≡1 mod q, n_q|p ⟹ n_q=1 normal.
FTGT: L/K finite Galois = finite+normal+separable; G=Gal(L/K), |G|=[L:K]; inclusion-reversing bijection fields↔subgroups, [L:M]=|H|, [M:K]=[G:H]; M/K normal ⟺ H◁G, Gal(M/K)≅G/H. x⁴−2: roots ±α,±iα (α=⁴√2), L=ℚ(α,i), [ℚ(α):ℚ]=4 (Eisenstein at 2), i∉ℝ ⟹ [L:ℚ]=8. r:α↦iα,i↦i (order 4); s:α↦α,i↦−i (order 2); srs⁻¹=r⁻¹ ⟹ G≅D₄. Lattice: G↔ℚ; ⟨r⟩↔ℚ(i), ⟨r²,s⟩↔ℚ(√2), ⟨r²,rs⟩↔ℚ(√−2); ⟨r²⟩↔ℚ(√2,i), ⟨s⟩↔ℚ(α), ⟨r²s⟩↔ℚ(iα), ⟨rs⟩↔ℚ((1+i)α), ⟨r³s⟩↔ℚ((1−i)α); non-central order-2 not normal (⟨s⟩↔ℚ(⁴√2) not Galois); {e}↔L.
Solvable = subnormal series with abelian quotients (equiv. derived series → {e}). Aₙ simple n≥5 (3-cycles generate; normal subgroup contains a 3-cycle then all). Nonabelian simple ⟹ not solvable (G'=G). Subgroup of solvable is solvable ⟹ Sₙ not solvable n≥5. A₄ not simple (Klein V normal), S₄ solvable. Radical-solvable ⟺ Gal solvable (Galois); forward via roots of unity + Kummer cyclic. General quintic: K=ℚ(t₁..t₅), Gal=S₅ ⟹ not solvable by radicals = Abel–Ruffini (Ruffini 1799 incomplete, Abel 1824). Caveat: x⁵−2 solvable (F₂₀=ℤ₅⋊ℤ₄, order 20); x⁵−x−1 has S₅.
Irreducible/prime and UFD defs. Existence via ACC: PID Noetherian; if a₀ has no factorization, reducible a₀=a₁b₁, one factor with no factorization, (a₀)⊊(a₁)⊊… violates ACC. (Without ACC fails: ring of all algebraic integers, 2=√2·√2=…) Uniqueness: in a PID irreducible⇒prime ((p,a)=(d), d unit or associate of p; p∤a ⟹ d unit ⟹ 1=px+ay ⟹ p|b); standard induction. UFD not PID: ℤ[x] (Gauss's lemma), (2,x) non-principal (even constant terms, 1∉(2,x)); also k[x,y]; obstruction Krull dim >1.
Over ℂ, G finite; χ_ρ(g)=tr ρ(g) class function, k=#classes. ⟨χ,ψ⟩=1/|G| Σ χ(g)ψ̄(g). Row: ⟨χ_i,χ_j⟩=δ_ij. Column: Σ_i χ_i(g)χ_i(h)̄ = |C_G(g)| if g~h else 0 (Frobenius; Schur underneath). S₃ classes {e},{transpositions}(3),{3-cycles}(2). Table: triv(1,1,1), sgn(1,−1,1), std(2,0,−1). Permutation rep ℂ³ char = fixed points (3,1,0). ⟨perm,triv⟩=1, ⟨perm,sgn⟩=0, ⟨perm,std⟩=1 ⟹ perm≅triv⊕std (3=1+2). Norm ⟨perm,perm⟩=2=1²+1².
vaiu-sci-math-prof-geometry; named only algebraic objects (torus ℤ², Klein ⟨a,b|abab⁻¹⟩=ℤ⋊ℤ, abelianization ℤ⊕ℤ/2); topology derivation is Geometry's.vaiu-cai-sec-prof-crypto + NIST PQC (ML-KEM/Kyber, ML-DSA/Dilithium); warned against roll-your-own; taught in-scope algebra (SVP/CVP, LWE Regev 2005, Ring-LWE Lyubashevsky–Peikert–Regev, cyclotomic ℤ[ζ_m], ideal lattices, Minkowski, NTT, worst-case-to-average-case).