An introduction to central simple algebras and their by Grégory Berhuy

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By Grégory Berhuy

Crucial easy algebras come up clearly in lots of components of arithmetic. they're heavily attached with ring idea, yet also are vital in illustration concept, algebraic geometry and quantity concept. lately, magnificent functions of the speculation of primary easy algebras have arisen within the context of coding for instant conversation. The exposition within the publication takes good thing about this serendipity, featuring an creation to the idea of relevant easy algebras intertwined with its functions to coding idea. Many effects or structures from the traditional conception are awarded in classical shape, yet with a spotlight on specific suggestions and examples, usually from coding conception. issues lined comprise quaternion algebras, splitting fields, the Skolem-Noether Theorem, the Brauer workforce, crossed items, cyclic algebras and algebras with a unitary involution. Code structures give the chance for plenty of examples and particular computations. This booklet presents an creation to the idea of primary algebras obtainable to graduate scholars, whereas additionally offering themes in coding thought for instant verbal exchange for a mathematical viewers. it's also appropriate for coding theorists attracted to studying how department algebras can be precious for coding in instant verbal exchange

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Let I be a minimal right ideal of A. Let us show first how (1) implies properties (2) − (4). Let M be a finitely generated simple right A-module. In particular, M is non-trivial and therefore M ∼ =A I n for some n ≥ 1 by (1). Since M is simple, we necessarily have n = 1. Otherwise I n , and thus M , would have a non-trivial submodule. Hence M∼ =A I; this proves (2). Assume now that M is a non-zero finitely generated A-module. If M is free, then M∼ =A An , where n = rkA (M ). Since M and An are isomorphic as k-vector spaces, comparing dimensions then shows that dimk (M ) = rkA (M ) dimk (A).

A direct computation shows that the map γ : (a, b)k −→ (a, b)k is k-linear and satisfies γ(q1 q2 ) = γ(q2 )γ(q1 ) for all q1 , q2 ∈ (a, b)k . 3. Let Q = (a, b)k be a quaternion k-algebra. The reduced norm of q = x + yi + zj + tij ∈ Q is defined by NrdQ (q) = x2 − ay 2 − bz 2 + abt2 ∈ k. One can easily check that NrdQ (q) = qγ(q) = γ(q)q. In particular, for all q1 , q2 ∈ (a, b)k , we have NrdQ (q1 q2 ) = NrdQ (q1 )NrdQ (q2 ). We now give an explicit criterion which permits to decide whether or not (a, b)k is a division algebra.

14. 6 we get A∼ = EndA (A) ∼ = EndA (I r ) ∼ = Mr (EndA (I)) = Mr (D). One may check that all these isomorphisms are k-linear, so we get the existence part of the theorem. 18 and the formula dimk (A) = r 2 dimk (D). 8 (3) that Z(D) ∼ =k Z(Mr (D)) ∼ =k Z(A). This completes the proof. 2. Let A and B be two central simple k-algebras. For every integer n ≥ 1, we have Mn (A) ∼ =k Mn (B) ⇔ A ∼ =k B. Proof. Assume that Mn (A) ∼ =k Mn (B). By Wedderburn’s theorem, we may write A∼ =k Ms (D ), where D, D are central division k-algebras.

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