# system of equations with coefficients in finite field

Suppose we have three simultaneously equations with $4$ variables with coefficients in finite fields, i.e.

$$\alpha_1A_1 + \beta_1B_1+\gamma_1C_1 + \theta_1D_1=x$$ $$\alpha_2A_1 + \beta_2B_1+\gamma_2C_1 +\theta_2D_1=y$$ $$\alpha_3A_1 + \beta_3B_1+\gamma_3C_1 +\theta_3D_1=z$$

Is it possible to solve for $A_1,B_1,C_1,D_1$ solely from equations above? Maybe we can use some properties of finite fields?

• You can solve such a system in the same way you solve systems of linear equations with real or complex coefficients, for example by transforming it to echelon form, which allows you to see whether there are solutions at all. Note that the algorithm to pass to the echelon form only reuires operations that are possible in every field. – Hagen Knaf Jul 10 '14 at 10:07

Specifically, you can never get a unique solution to $3$ equations in $4$ variables, even if the field is finite.