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so in college I am taking Linear Algebra and we're having a test. In this test we have to solve a system of equations over a finite field.

Let's have a finite field $Z_n = \{0, 1, ..., n - 1\}$. So for example a field $Z_3 = \{0, 1, 2\}$.

Now let's say that we have to solve the system of equations below in $Z_3$.

$x + 2y + z = 1 \\ x + 2y + 2z = 1 \\ 2x + y + z = 1$

As I am farily new to this I am not confident enough solving this. However I could not find any website which would provide practice problems (with solutions) like these.

Could you please suggest me where I'd find some?

Thank you

Jasasul
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  • Can't you just use your favourite method of elimination ... but here, if you add the last two equations you discover they are inconsistent as they reduce to $0=2$ - or was the second supposed to be $x+y+2z$ rather than $x+2y+2z$? – Mark Bennet Nov 06 '21 at 17:37
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    There's literally no meaningful difference between solving such equations over finite fields versus solving them over the reals. Every single step you'd do in the reals can be done the exact same way in finite fields. The only difference is in how you add, multiply and invert the numbers that pop up. – Vercassivelaunos Nov 06 '21 at 17:49
  • at the beginning, you may want to make a table of the multiplication group for your field, i.e. $\mathbb F_p -\big{0\big}$ that way you can 'cache' results and just look up products and their inverses when you are doing elimination – user8675309 Nov 06 '21 at 18:15
  • As long as the systems are linear, you can apply the usual methods you learned when studying linear algebra over the reals. The only difference is that this time the row operations use the arithmetic of the finite fields. Like here.

    BTW, $\Bbb{Z}_n$ is a field only when $n$ is a prime. There are other finite fields, but they are a bit more complicated, and usually constructed as quotient rings of polynomial rings over $\Bbb{Z}_p$.

     – Jyrki Lahtonen Nov 06 '21 at 18:25
  • This time you would run into an exceptional reduced row echelon form because the matrix of coefficients of this system is non-singular. Still, the usual technique applies. – Jyrki Lahtonen Nov 06 '21 at 18:27

2 Answers2

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A commonly taught method for solving systems of linear equations over a field is Gaussian elimination. It is a simple algorithm that works over any field, also finite fields.

Servaes
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$$x + 2y + z = 1 \\ x + 2y + 2z = 1 \\ 2x + y + z = 1$$

Add the first two equations and we get $$ 2x+y=2$$ Plug in the third equation we get $z=2 $

Subtracting the first equation from the second results in $z=0$

Therefore the system as given is inconsistent.

However if we change the middle equation to $$x+y+2z=1$$ then we have $$x=y=z=1$$

Mohammad Riazi-Kermani
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