Seeing that $\Bbb F_2[x]/(x^2+x+1)$ is a field

Question

I have some basic question with polynomials appreciate if someone could explain me this.

Following is additional and multiplication tables and it is say that this is a field. Have no idea why say it is field.

Please explain me.

Answer 1

Just note that, for multiplication, every non-zero element on the first column has, on its line, a $1$, which means it has a inverse to the right... Now, for every element on the first row, which contains nonzero elements, it has a 1 on its column, which means it is left-invertible.

Thus, it is a field since every nonzero element is invertible.

Answer 2

If $R$ is a ring with unity and $f\in R[x]$ a polynomial, then $R[x]/\langle f\rangle$ is always at least a ring. To be a field, it needs a unity (check) and each nonzero element must have a multiplicative inverse (check).