Results on group rings, cyclotomic local rings, character products and circulant Hadamard matrices, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• The norm equation — Lemma 2.1. A circulant sign matrix with first row h is Hadamard if and only if h h* = n in Z[C_n].
• Even squares — Corollary 2.2. The order of a circulant Hadamard matrix greater than 1 is an even square.
• Barker sequences of even length — Proposition 3.1. A Barker sequence of even length n > 2 is the first row of a circulant Hadamard matrix.
• Rings generated by roots of unity — Lemma 1.1. They are free abelian of finite rank and Noetherian, with Noetherian localizations, and B/pB is nonzero.
• Principal maximal ideals — Lemma 1.2. In a Noetherian local domain with maximal ideal (t), every nonzero element is a unit times a power of t.
• The unramified local ring — Lemma 2.1. Localizing Z[eta], eta of order prime to p, at a maximal ideal above p gives a local ring with maximal ideal pA.
• Adjoining a p-power root of unity — Lemma 3.1. A[xi] is free with basis 1, ..., xi^(D-1), local with maximal ideal (xi - 1), p = unit (xi-1)^D, and R(xi) is divisible by p^l exactly when all coefficients of R are.
• The order restriction (Turyn) — Proposition 4.1. A circulant Hadamard matrix of order n > 1 has n = 4u^2 with u odd.
• Character comparison (OpenAI) — Lemma 1.1. If F K = p^e b in A[C_(p^e)] with b a unit, then F(rho)/F(1) and K(rho)/K(1) are units with residue 1.
• Kronecker's criterion — Lemma 2.1. An element of a ring generated by roots of unity with all images of absolute value 1 is a root of unity; a root of unity with residue 1 in characteristic p has p-power order.
• Alternating products (OpenAI) — Proposition 3.1. For x x* = u^2 in Z[i][C_(u^2)], u > 1 odd, the alternating product of character values is a root of unity of odd order.
• First-order terms — Lemma 1.1. For t in the maximal ideal of a local domain, 1 + tL maps to L modulo the maximal ideal as a homomorphism.
• The circulant Hadamard conjecture (OpenAI) — Theorem 2.1. A real circulant Hadamard matrix of order n exists if and only if n is 1 or 4.
• Even-length Barker sequences — Corollary 3.1. A Barker sequence of even length n exists only for n = 2 and n = 4.

Reading order

Read the lessons in order. The first sets up the group ring and the Barker reduction. The second proves the local facts about cyclotomic integers and the order restriction. The third proves that the alternating products are roots of unity of odd order, and the fourth proves the theorem.