TITLE: Seeing that $\Bbb F_2[x]/(x^2+x+1)$ is a field
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: 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).