Extension Fields Generated by Elements

# Extension Fields Generated by Elements

 Definition: Let $(F, +, *)$ be a field and let $(K, +, *)$ be an extension field. Let $u_1, u_2, ..., u_n \in K$. The Extension Field of $F$ Generated by $u_1, u_2, ..., u_n$ is denoted $F(u_1, u_2, ..., u_n)$ and is defined to be the smallest extension field of $F$ containing $u_1, u_2, ..., u_n$. If we only consider a single element $u \in K$, then $F(u)$ is called a Simple Extension Field of $F$ Generated by $u$.

Consider a simple extension $F(u)$ where $u \in K$ is algebraic over $F$. If the degree of $u$ over $F$ is $n$, it can be shown that:

(1)
\begin{align} \quad F(u) = \{ a_0 + a_1u + ... + a_{n-1}u^{n-1} : a_0, a_1, ..., a_{n-1} \in F \} \end{align}

For example, for the field of rational numbers $\mathbb{Q}$ and the element $\sqrt{2} \in \mathbb{R}$ we have that:

(2)
\begin{align} \quad \mathbb{Q}(\sqrt{2}) = \{ a + b\sqrt{2} : a, b \in \mathbb{Q} \} \end{align}

We will now look at an important result regarding simple extension fields generated by algebraic elements over $F$.

 Theorem 1: Let $(F, +, *)$ be a field and let $(K, +, *)$ be a field extension. If $u \in K$ is algebraic over $F$ and $p \in F[x]$ is the minimal polynomial of $u$ over $F$ then $\displaystyle{K(u) \cong K[x] / \langle p(x) \rangle}$.
• Proof: Define a function $\phi : K[x] \to F$ for all $f \in K[x]$ by:
(3)
\begin{align} \quad \phi (f) = f(u) \end{align}
• Let $f, g \in K[x]$. Then:
(4)
\begin{align} \quad \phi (f + g) = (f + g)(u) = f(u) + g(u) = \phi (f) + \phi (g) \end{align}
(5)
\begin{align} \quad \phi (f * g) = (f * g)(u) = f(u) * g(u) = \phi (f) * \phi (g) \end{align}
(6)
\begin{align} \quad \phi (1) = 1(u) = 1 \end{align}
• So $\phi : K[x] \to F$ is a ring homomorphism. Now we have that:
(7)
\begin{align} \quad \ker (\phi) = \{ f \in F[x] : \phi (f) = 0 \} = \{ f \in F[x] : f(u) = 0 \ = \langle p(x) \rangle \end{align}
• We also see that:
(8)
\begin{align} \quad \phi (K[x]) = K(u) \end{align}
(9)
\begin{align} \quad K(u) \cong K[x] / \langle p(x) \rangle \quad \blacksquare \end{align}