Degree of Field Extensions is Multiplicative

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Theorem

Let $E, K$ and $F$ be fields.

Let $E / K$ and $K / F$ be finite field extensions.


Then $E / F$ is a finite field extension, and:

$\index E F = \index E K \index K F$

where $\index E F$ denotes the degree of $E / F$


Proof

First, note that $E / F$ is a field extension as $F \subseteq K \subseteq E$.

Suppose that $\index E K = m$ and $\index K F = n$.

Let $\alpha = \set {a_1, \ldots, a_m}$ be a basis of $E / K$, and $\beta = \set {b_1, \ldots, b_n}$ be a basis of $K / F$.

We wish to prove that the set:

$\gamma = \set {a_i b_j: 1 \le i \le m, 1 \le j \le n}$

is a basis of $E / F$.


As $\alpha$ is a basis of $E / K$, we have, for all $c \in E$:

$\ds c = \sum_{i \mathop = 1}^m c_i a_i$, for some $c_i \in K$.

Define $\ds b := \sum_{j \mathop = 1}^n b_j$ and $d_i := \dfrac {c_i} b$.

Note $b \ne 0$ since $\beta$ is linearly independent over $F$, and $d_i \in K$ since $b, c_i \in K$.


Now we have:

$\ds c = \sum_{i \mathop = 1}^m \frac {c_i} b \cdot b \cdot a_i = \sum_{i \mathop = 1}^m \sum_{j \mathop = 1}^n d_i a_i b_j$

Thus $\gamma$ is seen to be a spanning set of $E / F$.


To show $\gamma$ is linearly independent, suppose that for some $c_{i j} \in F$:

$\ds \sum_{i \mathop = 1}^m \sum_{j \mathop = 1}^n c_{i j} a_i b_j = 0$


Then we have (as fields are commutative rings):

\(\ds \sum_{i \mathop = 1}^m \sum_{j \mathop = 1}^n c_{i j} a_i b_j\) \(=\) \(\ds 0\)
\(\ds \leadstoandfrom \ \ \) \(\ds \forall i \in \set {1, \ldots, m}: \, \) \(\ds \sum_{j \mathop = 1}^n c_{i j} b_j\) \(=\) \(\ds 0\) Definition of Linearly Independent Set: $\alpha$ over $K$
\(\ds \leadstoandfrom \ \ \) \(\ds \forall i \in \set {1, \ldots, m}, j \in \set {1, \ldots, n}: \, \) \(\ds c_{i j}\) \(=\) \(\ds 0\) Definition of Linearly Independent Set: $\beta$ over $F$


Hence $\gamma$ is a linearly independent spanning set; thus it is a basis.


Recalling the definition of $\gamma$ as $\set {a_i b_j: 1 \le i \le m, 1 \le j \le n}$, we have:

$\size {\gamma} = m n = \index E K \index K F$

as was to be shown.

$\blacksquare$


Also known as

This result is also known as the degree equation, or the tower rule or tower law, from the definition of a tower of fields.


Sources

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