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Results tagged with finite-fields
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user 12858
A finite field is a field with a finite number of elements. For each prime power $q^k$, there is a unique (up to isomorphism) finite field with $q^k$ elements. Up to isomorphism, these are the only finite fields.
5
votes
Classifications of finite simple objects
Somewhat related to Igor Pak's comment is the classification of the finite irreducible Coxeter groups. Of course they are not "simple" as groups, but the irreducibility seems the natural replacement f …
- 4,713
7
votes
How to prove that a quaternion algebra over ℤₚ is isomorphic to Mat₂(ℤₚ) for p prime?
At the risk of being redundant and repeating earlier answers, let me mention that this is explicitly contained in the book "Elementary number theory, group theory and Ramanujan graphs" by Davidoff, Sa …
- 6,614
6
votes
Gcd of polynomials over a finite field
Here is another, more elementary approach, which works over any field, not just finite fields.
Assume that $n \geq m > 0$, suppose that $a,b \neq 0$, and let $c=b/a$. Then
$$\gcd(x^n-a, x^m-b) = \gcd …
- 6,614
4
votes
Orthogonal Groups over finite fields
I think it's worth adding that there is a very detailed analysis of the orthogonal groups over arbitrary fields (not just finite fields, and including characteristic 2) in Dieudonné's "La Géométrie de …
- 6,614