Timeline for Counting matrices over finite fields of a given order
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Sep 8, 2016 at 10:01 | comment | added | Fedor Petrov | @Mr.Kho total degree (not a number! above it is $m_1+2m_2$, not $m_1+m_2$) of linear and quadratic factors of a polynomial $g(x)\in \mathbb{F}_p[x]$ is a total number of roots of $g$ in $\mathbb{F}_{p^2}$. For $g(x)=x^t-1$ it equals gcd$(t,q^2-1)$ as may be seen by considering a primitive root $\alpha$ in $\mathbb{F}_{p^2}$ and checking which powers of $\alpha$, $\beta=\alpha^s$, satisfy $\beta^t=1$. | |
| Sep 8, 2016 at 8:11 | comment | added | Mr. Kho | in one of your answers above you say that "this is standard ...total degree of linear and quadratic factors is gcd(t,$q^2−1$)". Is there a book which explains how to arrive at this? | |
| Oct 28, 2015 at 11:11 | comment | added | Fedor Petrov | If characteristic polynomial is $X^2+X+1$, then determinant is 1. | |
| Oct 28, 2015 at 9:20 | comment | added | Mr. Kho | @ Fedor Petrov, checking the determinants of the elements (matrices) of order 3 in GL(2,GF(32)), I find that all elements have determinant equal to 1. Is this a mere coincidence? | |
| Oct 23, 2015 at 11:35 | comment | added | Fedor Petrov | For $t=11$ we see that $x^{11}-1$ has 1 linear multiple $x-1$ and 5 quadratic multiples, that is, $m_1=1$, $m_2=5$. Thus we should have $1+5\cdot 32\cdot 31=4961$ matrices for which $X^{11}=1$, out of them there is 1 matrix of order 1 and 4960 matrices of order 11. | |
| Oct 23, 2015 at 7:43 | comment | added | Mr. Kho | Also, is there a paper/book/any reference material where I can find this? | |
| Oct 23, 2015 at 7:41 | comment | added | Mr. Kho | @ Fedor Petrov, I have tried this and it works for t=3, however, over the same field, for t=11 and t=31, my computer results are different from those I compute by hand. Anything I am missing? | |
| Oct 21, 2015 at 7:40 | comment | added | Mr. Kho | many thanks for your answer it solves the problem. | |
| Oct 21, 2015 at 7:27 | vote | accept | Mr. Kho | ||
| Oct 21, 2015 at 7:38 | |||||
| Oct 20, 2015 at 13:49 | history | edited | Fedor Petrov | CC BY-SA 3.0 |
edited body
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| Oct 20, 2015 at 10:57 | history | edited | Fedor Petrov | CC BY-SA 3.0 |
added 1561 characters in body
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| Oct 20, 2015 at 10:03 | history | answered | Fedor Petrov | CC BY-SA 3.0 |