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### 2.8. Fields

In this section, we study some important properties of field extensions. We also give an introduction to Galois theory. Unless otherwise stated, the letters F, K and L stand for fields in this section.

#### 2.8.1. Properties of Field Extensions

We have seen that if FK is a field extension, then K is a vector space over F. This observation leads to the following very useful definitions.

##### Definition 2.54.
 For a field extension F ⊆ K, the cardinality of any F-basis of K is called the degree of the extension F ⊆ K and is denoted by [K : F]. If [K : F] is finite, K is called a finite extension of F. Otherwise, K is called an infinite extension of F.
##### Proposition 2.30.
 Let F ⊆ K ⊆ L be a tower of field extensions. Then [L : F] = [L : ...

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