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.

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

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. |

Let F ⊆ K ⊆ L be a tower of field extensions. Then [L : F] = [L : ... |

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