# Solving a system of equations over finite fields with integer coefficients

I want to solve a system of equations in $GF(2^4)$, with irreducible polynomial being $x^4 + x + 1$. The field is defined as follows:

```
K.<x>= GF(2^4, modulus=x^4+x+1)
```

Now suppose our equations are in the form $2u + 5v = 3$. Can I use this equation directly or should I substitute the polynomial equivalent of the numbers? That is, replace $2$ with $x$ and $5$ with $x^2 + 1$ and $3$ with $x + 1$ which gives us:

```
u*x + v*x^2 + v = x + 1
```

If so, this is a huge overhead for me. Is there an alternative way?