Galois theory is an important tool for the study of fields. The primary objects of study in Galois theory are automorphisms of fields.
Consider the field . Then the map given by is a field automorphism; that is, and , and is a bijection. Of course, the map given by is also a field automorphism. Both of these automorphisms are the identity automorphism on , a subfield of . It turns out that and are the only automorphisms of that fix . Furthermore, the automorphisms and form a group, called the Galois group of over .
We now define Galois groups more rigorously.
Let be a field extension. Then the set of field automorphisms of that fix form a group under composition. This group is called the Galois group of and is denoted .
One may wonder if the elements of are the only elements of fixed by every element of . It turns out that this is not always the case. For example, if and , then is the trivial group, so every element of is fixed by . If the elements of are the only elements of fixed by , then we say that is a Galois extension.
Many beautiful results can be obtained with a bit of Galois theory. For example, one can prove that it is impossible to trisect an angle using Galois theory.
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