# System of equation over GF(211) (corrected)

I have this system of equation. $a+b+c=171, a+2b+4c = 46, a+3b+9c = 170$.

My task is to solve this system over $GF(211)$. Is there any special process?

• What joriki says below. If there was a typo, and you are to work over the field $GF(223)$ (that does exist by virtue of 223 being a prime), then you can apply the usual linear algebra techniques and perform row operations on a suitably augmented matrix. See my answer for an example - it is over $GF(29)$. If there was now typo, and you are to find the solutions in the ring $\mathbb{Z}_{221}$, then you can try the same technique. But it may or may not work! It is possible that at some point you would need to divide by 13 or 17, and that's a no-no. – Jyrki Lahtonen Nov 25 '12 at 8:29
• (cont'd) but your matrix of coefficients is of Vandermonde type, so it looks like the process should work (the determinant is not divisible by either 13 or 17). In more general situations you may not get a unique solution, and things become more complicated. – Jyrki Lahtonen Nov 25 '12 at 8:34

There is no such thing as $GF(221)$; finite fields exist only for prime-power orders; $221=13\cdot17$.