An introduction to central simple algebras and their by Grégory Berhuy

By Grégory Berhuy

Valuable basic algebras come up clearly in lots of parts of arithmetic. they're heavily hooked up with ring conception, yet also are very important in illustration concept, algebraic geometry and quantity concept. lately, unbelievable purposes of the speculation of valuable easy algebras have arisen within the context of coding for instant verbal exchange. The exposition within the booklet takes benefit of this serendipity, featuring an creation to the speculation of significant basic algebras intertwined with its purposes to coding conception. Many effects or structures from the normal conception are provided in classical shape, yet with a spotlight on specific concepts and examples, frequently from coding thought. 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 buildings give the chance for plenty of examples and particular computations. This publication presents an advent to the speculation of vital algebras obtainable to graduate scholars, whereas additionally featuring issues in coding conception for instant conversation for a mathematical viewers. it's also appropriate for coding theorists drawn to studying how department algebras can be priceless for coding in instant verbal exchange

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For 1 ≤ i ≤ n, we have canonical projections πi : M n −→ M m = (mj )1≤j≤n −→ mi and canonical injections ιi : M −→ M n m −→ (0, . . , 0, m, 0, . . , 0). Let f : M −→ M be an endomorphism. For 1 ≤ i ≤ n, let fi = πi ◦ f . Then fi : M n −→ M is R-linear and for all m ∈ M n , we have n n f (m) = (f1 (m), . . , fn (m)). Now observe that fi (m) = fi (ι1 (m) + · · · + ιn (m)) = fi (ι1 (m)) + · · · + fi (ιn (m)). Putting things together, we finally get n n πi ◦ f ◦ ι1 , . . , f= i=1 πi ◦ f ◦ ιn . i=1 The idea of the proof is that, according to the formula above, f should be completely determined by the maps πi ◦ f ◦ ιj .

Let us show that Q = (i, 1 + 2i)Q(i) is a division algebra. Assume to the contrary that Q is split. 5, we have 1 + 2i = NQ(i)(ζ8 )/Q(i) (ξ) for some ξ ∈ Q(i)(ζ8 )× , where ζ8 denotes a primitive 8-th root of 1. As in the previous example, this implies that (1 + 2i)z 2 = x2 − iy 2 , for some x, y, z ∈ Z[i]. 1. PROPERTIES OF QUATERNION ALGEBRAS 25 Since Z[i] is a unique factorization domain, one may assume that x, y, z are coprime, and show as before that 1 + 2i y, using that 1 + 2i is an irreducible element of Z[i].

This contradicts the minimality of m. Hence m = 1, so I contains an element of the form 1 ⊗ b. Since B is simple, arguing as at the beginning of the proof shows that I contains 1 ⊗ 1, so I = A ⊗k B and we are done. 7. This result is not true if A is not central. For example, C is a simple R-algebra. However, we have ∼R C × C. C ⊗R C = Since C × {0} is a non-trivial ideal of C × C, it follows that C ⊗R C is not simple. 8. Let A and B be two k-algebras, and let L/k be a field extension. Then the following properties hold: (1) if A and B are central simple, so is A ⊗k B; (2) A is central simple over k if and only if A ⊗k L is central simple over L.

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