Brauer Group

Central simple algebras, Brauer equivalence, and Br(ℝ) ≅ ℤ/2ℤ

Central Simple Algebras over ℝ

ℝ — Real numbers

The base field. Identity element of Br(ℝ).

dim=1, center=ℝ, division algebra

ℍ — Quaternion algebra

i²=j²=−1, ij=k. Non-commutative. Unique ℝ-division algebra besides ℝ,ℂ (Frobenius).

dim=4, center=ℝ, [ℍ]≠0 in Br(ℝ)

M₂(ℝ) — 2×2 matrices

Simple but not division. Brauer-equivalent to ℝ (Morita). [M₂(ℝ)]=[ℝ]=0.

dim=4, center=ℝ, [M₂(ℝ)]=0 in Br(ℝ)

ℂ — Complex numbers

Not central simple over ℝ (center=ℂ≠ℝ). Extension, not in Br(ℝ).

dim=2, NOT in Br(ℝ), lives in Br(ℂ/ℝ)

M₂(ℍ) — 2×2 quaternion matrices

Brauer-equivalent to ℍ. [M₂(ℍ)]=[ℍ] in Br(ℝ).

dim=16, [M₂(ℍ)]=[ℍ] (same class)

Br(ℝ) ≅ ℤ/2ℤ — Group law (tensor product)

[ℝ] = 0 [ℍ] = 1
[ℍ] ⊗ [ℍ] = [ℍ⊗ℍ] = [M₄(ℝ)] = [ℝ] = 0
So [ℍ] has order 2 in Br(ℝ).
Brauer group Br(k): equivalence classes [A] of central simple k-algebras, where A~B if A⊗Mₙ(k)≅B⊗Mₘ(k). Group operation: tensor product ⊗_k.

Key results:
Br(ℝ) ≅ ℤ/2ℤ = {[ℝ],[ℍ]}
Br(ℂ) = 0 (algebraically closed fields have trivial Brauer group)
Br(𝔽_q) = 0 (finite fields, Wedderburn's theorem)
Br(ℚ_p) ≅ ℚ/ℤ (local fields)

Frobenius 1877: Only finite-dim ℝ-division algebras are ℝ, ℂ, ℍ.