Example. complex and real numbers, rationals and polynomials [1204]

    1. $\mathbb{C}$ is a field extension of $\mathbb{R}$ and $[\mathbb{C} : \mathbb{R}] = 2$.
    2. $\mathbb{R}$ is a field extension of $\mathbb{Q}$ and $[\mathbb{R} : \mathbb{Q}] = \infty$.
    3. Let $F$ be a field and $p(x)$ be an irreducible polynomial of degree $\geq 1$ in $F[x]$. Then $K = F[x] / (p(x))$ is a field extension of $F$ and $[K : F] = \deg p(x)$.
    4. $F = \mathbb{Q}, p(x) = x^{2} - 2, K = \mathbb{Q}[x] / (x^{2} - 2)$. Then $K \cong \mathbb{Q}[\sqrt{ 2 }] = \{ a + b\sqrt{ 2 } \mid a, b \in \mathbb{Q} \}$.
    5. $[\mathbb{F}_{p}(t), \mathbb{F}_{p}] = \infty$. $1, t, t^{2}, \dots, t^{n}, \dots$ are linearly independent over $\mathbb{F}_{p}$, so $\dim_{\mathbb{F}_{p}}\mathbb{F}_{p}(t) = \infty$.