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 Feb 4 comment What is the number of $0\leq j_1, j_2, \ldots, j_n\leq v-1$ such that $\sum_{i=1}^{n}j_i\equiv 0 \mod v$? Yup, should be quite easy. Maybe change a bit for n odd or even.. Feb 3 comment What is the number of $0\leq j_1, j_2, \ldots, j_n\leq v-1$ such that $\sum_{i=1}^{n}j_i\equiv 0 \mod v$? Thanks! Using your answer here it was fairly simple to get a recursive formula: If $N_n$ is the number question (with the $j$'s being $>0$, then it seems that $N_n=(v-1)^{n-1} - N_{n-1}$. I did it a bit quickly but it should be right. From that I think I'll be able to get a formula. Feb 3 comment What is the number of $0\leq j_1, j_2, \ldots, j_n\leq v-1$ such that $\sum_{i=1}^{n}j_i\equiv 0 \mod v$? So, I was not careful again. In my original question I needed to put that $1 \leq \j_1, j_2, \ldots, j_n\leq v-1$. I accepted your answer because it is correct for the question I asked and may prove to be helpful to others. On top of that, I think I can work on a proof by using your idea. In particular, for the case $n=3$, we have $v^2$ tuples including the cases when any of $j_1,j_2,j_3$ are $0$, and then we subtract the cases that $j_1+j_2\equiv 0 \bmod v$. Then we end up having (v-1)(v-2). I think this should be an easy application of the inclusion-exclusion principle.. I'll work on it. Feb 3 comment What is the number of $0\leq j_1, j_2, \ldots, j_n\leq v-1$ such that $\sum_{i=1}^{n}j_i\equiv 0 \mod v$? @David Ooops, I meant for the case $n=3$. Corrected now, thank you. Nov 16 comment Finite fields: factorization of the trace function over the base field Jyrki and @Thomas: Thanks both of you for providing an answer. Jyrki, you have been answering a lot of my finite fields questions, thanks for all the help. Nov 12 comment Finite fields: factorization of the trace function over the base field @ThomasAndrews Your comments are all valid.. I'll modify the question to make more sense. Apr 21 comment How can I construct the matrix representing a linear transformation of a 2x2 matrix to its transpose with respect to a given set of bases? @ahorn That's a good point, Indeed my choice was arbitrary. I edited my answer to reflect this. Mar 16 comment Finite field question involving the trace and a permutation. Many thanks Jyrki, I'll need to study this a bit. I'll update later. Mar 16 comment Finite field question involving the trace and a permutation. I know a couple of people who were claiming that they could do math better after a couple of beers :) Anyway, I think I have an idea, but I have to double think before posting anything, I'm not sure yet. Mar 10 comment Bijections of a finite field that preserve the kernel of the trace I have realized what my $f(x)$ should be. Indeed, I was asking too much. If you want, check my updated question here: math.stackexchange.com/questions/1183155/… Feb 24 comment Bijections of a finite field that preserve the kernel of the trace Thank you for your clear and thorough answer, I was quite off with my conjecture about these monomials. I will have to think how/if the question can be restated to suggest some other class of polynomials. Feb 24 comment Bijections of a finite field that preserve the kernel of the trace I think I was being quite careless when I wrote that title. It should be just bijections. Jul 7 comment Which powers of a primitive element of a finite field yield a generator of a finite field extension? Thanks for your answer. Now, you say that WLOG, $i|(q^m-1)$, else $i$ is associate to a divisor's residue in $\Bbb{Z}_{q^m-1}$. Could you comment on that? I fail to understand it.. Jul 4 comment Which powers of a primitive element of a finite field yield a generator of a finite field extension? Note that the last bullet is equivalent to "there is no $s$ such that \$0