Below is example of extending Field $F$ to include root of a irreducible polynomial. After extending Field $F$ to ploynomial Field $F[t]$ , we take polynomial $p=t^{3}+t+1$ and declare it is equal to zero in that polynomial ring. It makes
$t^{3}+t+1=0$ so it leaves
$t^{3}+t^{2}+t+1=t^{2}$
After all the subsequent subsitution it will leave only the first,second and third row. My question is
How the other elements of row 3 are eleminated. I only understand the above two subsitution. How will the further subsitution works?
Roots of this polynomial is $t$, how?
Is the polynomial is minimal polynomial?
