Abstract Model of Algebraic Extensions

Theorem
Let $K$ be a field and $\alpha \in \overline{K}$ be an element of the algebraic closure of $K$ which is algebraic over $K$.

Let $m_\alpha$ be the minimal polynomial of $\alpha$ over $K$.

Then
 * $K[\alpha]\cong K[x]/\langle m_\alpha\rangle$.

Proof
Let $\phi:K[x]\mapsto K[\alpha]$ be the homomorphism which fixes $K$ and maps $x$ to $\alpha$.

The kernel of $phi$ consists of all polynomials which vanish at $\alpha$, which is precisely the ideal $\langle m_\alpha\rangle$.

Thus, our result follows by the First Fundamental Theorem on Ring Homomorphisms.