# An introduction to central simple algebras and their by Grégory Berhuy

By Grégory Berhuy

Primary uncomplicated algebras come up evidently in lots of components of arithmetic. they're heavily attached with ring thought, yet also are vital in illustration concept, algebraic geometry and quantity thought. lately, spectacular purposes of the speculation of imperative basic algebras have arisen within the context of coding for instant conversation. The exposition within the ebook takes good thing about this serendipity, featuring an advent to the idea of imperative uncomplicated algebras intertwined with its functions to coding concept. Many effects or structures from the traditional conception are awarded in classical shape, yet with a spotlight on particular strategies and examples, frequently from coding concept. issues lined comprise quaternion algebras, splitting fields, the Skolem-Noether Theorem, the Brauer staff, crossed items, cyclic algebras and algebras with a unitary involution. Code buildings give the chance for plenty of examples and particular computations. This booklet presents an creation to the speculation of important algebras obtainable to graduate scholars, whereas additionally featuring themes in coding idea for instant conversation for a mathematical viewers. it's also appropriate for coding theorists drawn to studying how department algebras might be worthy for coding in instant conversation

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**Example text**

Then the following properties hold: (1) every left (resp. right) A-module M has a natural structure of a k-vector space, induced by scalar multiplication. Moreover, M is ﬁnite dimensional over k whenever it is ﬁnitely generated as an A-module; (2) A has a minimal left (resp. right) ideal; (3) for all n, m ≥ 1, we have An ∼ =A Am ⇐⇒ n = m. Proof. The ﬁrst point is obvious. To prove (2), observe that any left (resp. right) ideal of A is in particular a linear subspace of A, hence ﬁnite dimensional over k since A is.

Proof. 8, so D contains R. Notice also that Z(D) is a ﬁeld containing R, so Z(D) is isomorphic to either R or C, since Z(D)/R has ﬁnite degree by assumption. If D is commutative, then D is a ﬁeld extension of R, hence is isomorphic to R or C. Assume now that D is not commutative, and let k = Z(D). Then k cannot be isomorphic to C. Otherwise, for all d ∈ D, the minimal polynomial μd,k of d would be of degree 1, meaning that d ∈ k. We then would have that D = k, contradicting the fact that D is not commutative.

Since L is a simple k-subalgebra of A and ZA (L) = L (since L is commutative), the last part of the Centralizer Theorem gives ∼L Mr (L ⊗L CA (L)). A ⊗k L = Since L ⊗L CA (L) ∼ =L CA (L), we are done. Exercises 1. Let A be a central simple k-algebra. Show that the group of all k-algebra automorphisms of A is canonically isomorphic to A× /k× . 2. Let A be a simple k-algebra, not necessarily central. Is every automorphism of A inner? 3. The goal of this exercise is to provide an elementary and totally explicit proof of Skolem-Noether’s theorem for matrix algebras.