By Eric Friedlander, M. R. Stein
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Extra info for Algebraic K-Theory. Proc. conf. Evanston, 1980
Then charðKÞ is a prime number p. Hence K can be considered as a finite dimensional space over the prime subfield F0 ﬃ Zp , and as such is isomorphic to F0n for some n 2 N. But then jKj ¼ jF0n j ¼ pn . We have thereby found out that the cardinality of a finite field can only be pn with p prime and n 2 N. This assertion has a sort of a converse, which, however, is less obvious: For every prime p and n 2 N there exists an (up to isomorphism) unique field having pn elements. An element a in an extension field K of F is said to be a root of the polynomial pﬃﬃﬃ f ðωÞ 2 F½ω if f ðaÞ ¼ 0.
1) can be formed over any field F, not only over R. However, if F = C, then this is not a division algebra. 2) we see that it is a division algebra if and only if for all αi ∈ F, α02 + α12 + α22 + α32 = 0 implies α0 = α1 = α2 = α3 = 0. This condition is of course fulfilled in R and some other fields (say, in Q), but not in algebraically closed fields. Still, one might wonder what algebra we get if F = C. 72. 3 Simple Rings Our next main goal is a theorem on finite division rings. In the next sections we shall progress slowly towards its proof, making several digressions that will turn out to be important later when dealing with more general rings and algebras.
3 Simple Rings Our next main goal is a theorem on finite division rings. In the next sections we shall progress slowly towards its proof, making several digressions that will turn out to be important later when dealing with more general rings and algebras. For a while division algebras will mostly lie dormant in our exposition. This section is devoted to the following notion. 8 A ring R is said to be simple if R2 = 0 and 0 and R are the only ideals of R. , xy = 0 for some x, y ∈ R, is needed to exclude pathological cases.
Algebraic K-Theory. Proc. conf. Evanston, 1980 by Eric Friedlander, M. R. Stein