Let $K$ be a field and $p(x) \in K[x]$ a monic irreducible polynomial of degree $n$, suppose $F/K$ is a field extension, and there exist $u \in K[x]$ which is a root of $p(x)$.

1. Let $K(u)$ be the smallest subfield of $F$ containing both $K$ and $u$, prove that $K(u) \cong K[x]/(p)$

2. Deduce that $K(u)=\{a_0+a_1u+....+a_{n-1}u^{n-1},a_0,a_1....\in K \}$

3)Compute $u^{-1} \in K(u)$

I guess that first one can be shown by fundamental theorem of homomorphism, and the second one by division algorithm,but how to do the third one ?

Let $K$ be a field and $p(x) \in K[x]$ a monic irreducible polynomial of degree $n$. Suppose $F/K$ is a field extension, and there exists $u \in F$ which is a root of $p(x)$.

1. Let $K(u)$ be the smallest subfield of $F$ containing both $K$ and $u$. Prove that $K(u) \cong K[x]/(p)$.

2. Deduce that $K(u)=\{a_0+a_1u+\cdots+a_{n-1}u^{n-1}:a_0,a_1,...a_{n-1}\in K \}$

3. Compute $u^{-1} \in K(u)$.

I guess that first one can be shown by fundamental theorem of homomorphism, and the second one by division algorithm, but how to do the third one ?