Let $F$ be a field and $P(X)\in F[X]$ a polynomial of degree $d\ge 1$. Describe the splitting fields of $F$-algebra $A=F[X]/(P(X))$.
My try: For every $a\in A$, there exists $f(X), g(X)\in F[X]$, $\deg g(X)<d$, such that $$ a = f(X)+(P(X)) = g(X)+(P(X)).$$ Could you please tell me how to connect an element in $A\otimes_F K$ with a matrix in $M_n(K)$? Thanks.