Is it possible to solve system of linear equations over finite fields using Wolfram Alpha? If yes, how can I do that?
Let us take a system $x+y+z=0$, $2x+y+2z=0$, $x+3y+z=0$.
If I want to solve this linear system over $\mathbb Z_7=\mathbb Z/(7)$, one thing I can do is to type
(x+y+z) mod 7=0, (2x+y+2z) mod 7=0, (x+3y+z) mod 7=0
into WA, here is a link. (This approach was suggested in an answer to this question.)
WA can solve this, but a solution is given by enumerating the residue classes. I would prefer some more compact notation, for example something similar like what WA does for the same system over real numbers, i.e., when I type x+y+z=0, 2x+y+2z=0, x+3y+z=0 into WA. (Getting a basis for a solution space would be nice, but even expression in the form given in this case seems somehow better than listing all residue classes.)
- Would I be able to solve system linear equations over finite fields $G(p^n)$? (For example if I chose the representation $GF(4)=\mathbb Z_2/(x^2+x+1)$ and tried to solve the above system in this field?)
- Is it possible to obtain some kind of more compact notation at least in the case of fields of the form $\mathbb Z_p=\mathbb Z/(p)$ (where $p$ is some prime number)?