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By Steven Dale Cutkosky

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Extra info for An Introduction to Galois Theory [Lecture notes]

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Quaternion algebras based codes In this section, we present two code constructions, one based on Hamilton quaternions and the other on generalized quaternion algebras. 3 for the coding motivation and related definitions. M. Alamouti [1] published a simple code construction for a wireless system with 2 transmit antennas, given as follows. Let x0 , x1 be two complex numbers that represent the information symbols to be sent, and let the codebook C be given by the following set of matrices C= x0 x1 −x1 x0 | x0 , x1 ∈ C .

We now prove the converse, which is known as Wedderburn’s theorem. 1 (Wedderburn’s Theorem). Any simple k-algebra A is isomorphic to Mr (D), for some integer r ≥ 1 and some division k-algebra D whose center is isomorphic to the center of A. In particular, a central simple algebra is isomorphic to a matrix algebra over a central division k-algebra. Moreover, the integer r and the isomorphism class of D only depend on the isomorphism class of A. More precisely, if I is a minimal right ideal of A, we have D∼ =k EndA (I) and A ∼ =A I r .

Let A and B be two k-algebras, and let L/k be a field extension. Then the following properties hold: (1) A ⊗k B is central over k if and only if A and B are central over k; (2) A ⊗k L is central over L if and only if A is central over k. Proof. 4 shows that Z(A ⊗k B) = Z(A) ⊗k Z(B). Therefore, dimk (Z(A ⊗k B)) = dimk (Z(A)) dimk (Z(B)). Now A ⊗k B is central over k if and only if dimk (Z(A ⊗k B)) = 1. The previous equality shows that it is equivalent to dimk (Z(A)) = dimk (Z(B)) = 1, that is A and B are central over k.

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