6
votes
2answers
187 views

A resemblance between 2 binomial identities - why?

Let $F$ be any field (or even ring). The following formal power series identity (i.e., equality in $F[[x]]$) holds for any $j \ge 0$: $$(1-x)^{-j} = \sum_{i \ge 0} \binom{i +j -1}{i} x^i $$ The ...
0
votes
2answers
87 views

Binomial formula in $GF(2^m)$

there is a binomial formula: $$(x+y)^n=\displaystyle\sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$$ When operations are done in $GF(2^m)$ then all positive integers are reduced $\bmod2$, so binomial formula ...
1
vote
1answer
310 views

Numbers of vectors in a vector space over a finite field, with different multiplication

I had a recent question in an assignment that I couldn't complete. We are given the following: $q$ is an odd prime power. $(F,+,\cdot)=\text{GF}\left(q^2\right)$. $K$ is the $q$ element subfield of ...