Lecture. Field extensions [l1]

  • Mar 30, 2026

Definition 1. field extension [1201]

    Let $F$ be a field. A field containing $F$ is called a field extensions of $F$ or an extension of $F$.

    Notation. some notations [1201A]

      We use

      $$ K / F \qquad \text{or} \qquad \begin{aligned} &K \\ &| \\ &F \end{aligned} $$

      to mean that $K$ is a field extension of $F$.

      When considering field extensions of $F$, the field $F$ is sometimes called the base field.

      Example 2. examples from characteristics [1202]

        • Every field of characteristic $0$ is a field extension of $\mathbb{Q}$.
        • Every field of characteristic $p$ is a field extension of $\mathbb{F}_{p}$.

        Definition 3. degree of field extensions [1203]

          3.1. degree [1203A]

            If $K$ is a field extension of $F$, then $K$ can be viewed as an $F$-vector space.

            Exegesis. vector space [1203AA]

              TODO

              The degree of $K / F$ is $$ [K : F] := \dim_{F}K .$$

              3.2. finiteness [1203B]

                A field extension $K / F$ is called finite if $[K : F] < \infty$.

                Example 4. 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 $\dim_{\mathbb{F}_{p}}\mathbb{F}_{p}(t) = \infty$.

                  Remark 5. [1205]

                    Theorem 6. [1206]

                      Let $F$ be a field and $p(x)$ be an irreducible polynomial in $F[x]$. Then

                      1. $K = F[x] / (p(x))$ is a field extension of $F$,
                      2. $\theta = \overline{x} \in K$ is a root of $p(x)$ in $K$,
                      3. $[K : F] = \deg p(x)$. Let $n = \deg p(x)$. Then $$1, \theta, \theta^{2}, \dots, \theta^{n-1}$$ is a basis of $K$ as an $F$-vector space. (Hence, $K = \{ a_{0} + a_{1} \theta + \cdots + a_{n-1}\theta^{n-1} \mid a_{0}, \dots, a_{n-1} \in F\}$.)

                      Proof. [1206A]

                        TODO

                        Theorem 7. [1207]

                          Let $F \subseteq K \subseteq L$ be fields. Then $[L:F]$ is finite if and only if $[L:K]$ and $[K:F]$ are both finite, and $$ [L:F] = [L : K][K : F]. $$

                          Proof. sketch [1207A]

                            $\implies$: Given basis of $L$ over $K$ and $K$ over $F$, denote by $\alpha_{i}, 1 \leq i \leq n$ and $\beta_{j}, 1 \leq j \leq m$ respectively, then we can prove $\alpha_{i}\beta_{j} \in L, \forall i, j$ are linearly independent over $F$. It follows that $[L : F] \geq mn$. So if $[L : F] < \infty$, it must follow that $m < \infty$ and $n < \infty$.

                            $\impliedby$: Assume now $m, n < \infty$. We can prove that $\alpha_{i}\beta_{j} \forall i, j$ actually spans $L$ over $F$ by showing that every element of $L$ can be written in the form $\sum F \alpha_{i} \beta_{j}$. Then $\alpha_{i} \beta_{j}$ is a basis of $L$ over $F$, hence $[L : F] = mn < \infty$.

                            When $[L : F] = \infty$, at most one of $[L:K]$ and $[K:F]$ can be finite. For example, $[\mathbb{C} : \mathbb{Q}] = \infty$ and $[\mathbb{R} : \mathbb{Q}] = \infty$ while $[\mathbb{C} : \mathbb{R}] = 2 < \infty$.