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Central simple algebras, the Brauer group, and group algebras

Pull the division rings over a fixed field K into a group. Central simple algebras, tensor products, the Brauer group, and a closing look at where group algebras send us next.

Central simple algebras

Fix a field K. A central simple algebra (CSA) over K is a finite-dimensional K-algebra A that is simple as a ring and has center exactly K (no bigger). By Artin-Wedderburn a CSA is M_n(D) for a division ring D whose center is also K. So classifying CSAs over K is, up to matrix size, the same as classifying the central division algebras over K — the genuinely noncommutative field-like objects sitting over K.

The Brauer group

Two CSAs are Brauer equivalent if they have isomorphic underlying division rings — equivalently A ≅ M_m(D) and B ≅ M_n(D) for the same D. The tensor product over K turns these classes into a group: the Brauer group Br(K). The product is [A]·[B] = [A ⊗_K B], the identity is [K] (the class of all matrix rings), and the inverse of [A] is the class of the opposite algebra [A^op], because A ⊗_K A^op ≅ M_d(K). The whole zoo of noncommutative division algebras over K is repackaged as one abelian group.

Key tensor identities over K (all CSAs):
  M_m(K) ⊗_K M_n(K)  ≅  M_{mn}(K)         (matrix rings collapse)
  A ⊗_K A^op         ≅  M_d(K),  d = dim_K A   (gives inverses)

Worked instance over K = R, the quaternions H:
  dim_R H = 4,  H^op ≅ H  (conjugation q -> q* is an anti-automorphism)
  so   H ⊗_R H  ≅  H ⊗_R H^op  ≅  M_4(R).
Hence in Br(R):   [H] + [H] = 0,   so  [H]  has order 2.

Result:  Br(R) = Z/2Z = { [R], [H] }.   Two classes, nothing more.
Computing Br(R) = Z/2Z: H tensored with itself is a matrix ring, so [H] is its own inverse.

That computation, Br(R) = Z/2Z, is the smallest nontrivial example and worth carrying around. Br(C) = 0 (algebraically closed), Br(F_q) = 0 (Wedderburn's little theorem in disguise — no noncommutative finite division rings), while Br(Q_p) ≅ Q/Z and Br(Q) sits inside a famous exact sequence of class field theory. The Brauer group is also identified with second Galois cohomology H^2(K, K̄*), which is where this thread joins the cohomology track.

Group algebras and where to go next

We close by tying back to a ring you already know is noncommutative in general: the group algebra K[G]. By Maschke it is semisimple when char K ∤ |G|, so Artin-Wedderburn applies and K[G] becomes a product of matrix rings — that product is the representation theory of G. Over C the blocks are M_{d_i}(C) and the d_i are the dimensions of the irreducibles, recovering the character table count Σ d_i^2 = |G|.

That is the arc of the whole track: noncommutativity does not destroy structure, it reorganizes it. Order-sensitivity gives birth to division rings; Schur and Wedderburn tame the semisimple case into matrix rings; the radical measures the rest; and the Brauer group organizes the leftover division algebras into something an abelian group can hold. The intuition you built over fields was never wrong — it just needed a left and a right.